8.G.A.5: Angle Facts for Triangles and Parallel Lines, and Angle-Angle Similarity
In plain English: 8.G.A.5 is the Common Core grade 8 math standard that asks students to use informal arguments to explain angle facts: the angles of a triangle add up to 180°, an exterior angle equals the sum of the two far interior angles, parallel lines cut by a transversal make equal angle pairs, and two triangles with two equal angles are similar. It prepares students for proofs in high school geometry.
Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.
Common Core State Standards for Mathematics · Domain: Geometry (G) · Cluster: Understand congruence and similarity using physical models, transparencies, or geometry software. Also written as 8.G.5 · Official standard
In grade 7, students used angle facts at a single point: vertical angles (the angles across from each other where two lines cross) are equal, and angles that form a straight line add up to 180°. This lesson moves those facts to new figures and asks students to explain why they are true. The explanations are informal arguments: convincing reasons built from cutting, tracing, sliding and turning figures, and from facts students already know, without the formal two-column proofs of high school.
The lesson starts with a transversal, a line that crosses two other lines. When the two lines are parallel, sliding one crossing along the transversal lands it on the other, so corresponding angles (angles in the same position at the two crossings) are equal. The pairs between the two lines, alternate interior angles (on opposite sides of the transversal) and same-side interior angles, follow from that. That fact explains why the three angles of any triangle add up to 180°, which the official example shows with three copies of one triangle. The same facts give the exterior angle rule and the angle-angle (AA) criterion: two triangles with two pairs of equal angles are similar (the same shape, possibly a different size).
Learning Objectives
By the end of this lesson, students will be able to:
Explain, with a translation (slide) along the transversal, why corresponding angles are equal when parallel lines are cut by a transversal, and use it to find alternate interior and same-side interior angles
Explain why the angles of any triangle add up to 180°, using three copies of a triangle or a line drawn parallel to one side
Explain why an exterior angle of a triangle equals the sum of the two interior angles that are not next to it
Explain why two triangles with two pairs of equal angles are similar, and use this to find unknown angles and lengths
Prior Knowledge Required
Students should already be comfortable with:
Vertical angles are equal, and angles that form a straight line (a linear pair) add up to 180° 7.G.B.5
Translations and rotations keep angle measures the same and take a line to a parallel line 8.G.A.1
Similar figures: one can be turned into the other by rigid motions and a dilation (a resizing from a center point), so matching angles are equal and matching sides share one scale factor 8.G.A.4
Solving linear equations with the variable on both sides 8.EE.C.7
Each student draws a large triangle of any shape on scrap paper, cuts it out, and marks its three corners with dots.
Warm-Up Prompt
"Tear off the three corners of your triangle. Put the three corners together so their points meet at one spot and their edges touch without gaps or overlaps. What do you notice about the outside edges? Compare with a neighbor whose triangle has a different shape."
Students see that the three corners always fill a straight line, which is a straight angle of 180°. Collect a few very different triangles (long and thin, nearly right, nearly equilateral) to show it is not a coincidence. Then ask: "Tearing shows it works for these triangles. How could we be sure it works for every triangle?" The lesson builds that reason.
Direct Instruction20 minutes
Build each fact on the board with a sketch, and have students copy the words next to it:
Transversal: a line that crosses two or more other lines. In Diagram 2, line m is a transversal of lines p and q. It makes four angles at each crossing, eight in all.
Corresponding angles sit in the same position at the two crossings, such as ∠2 and ∠6 (both above the line and to the right of m). Why they are equal: if p and q are parallel, a translation (slide) along m moves the top crossing exactly onto the bottom one. Line p lands on line q, and a slide keeps every angle measure, so each angle lands on its corresponding angle.
Alternate interior angles are between the two lines, on opposite sides of the transversal, such as ∠3 and ∠6. They are equal because ∠3 equals its vertical angle ∠2, and ∠2 equals ∠6 (corresponding).
Same-side interior angles are between the two lines, on the same side of the transversal, such as ∠4 and ∠6. They add up to 180°, because ∠4 and ∠2 form a straight line and ∠2 equals ∠6.
Triangle angle sum: arrange three copies of one triangle as in Diagram 1. The top line through C and D is parallel to the bottom line, and the transversal facts show that the three angles at B are a, b and c. They fill the straight angle along line AE, so a + b + c = 180° for every triangle.
Exterior angle: the angle between one side of a triangle and the extension of the next side, such as angle x in Diagram 2. The two remote interior angles are the triangle's angles that are not next to it. The exterior angle and the interior angle next to it make 180°, and so do all three interior angles, so the exterior angle equals the sum of the two remote interior angles.
Angle-angle (AA) criterion: if two angles of one triangle equal two angles of another, the triangles are similar. The third angles must match too, because both sets add up to 180°. Place the smaller triangle inside the larger one so that one pair of equal angles overlaps. The sides across from that angle meet the other sides at equal corresponding angles, so they are parallel (two lines that make equal corresponding angles with a transversal are parallel), and a dilation from the shared corner stretches the small triangle onto the large one.
The official example: three copies of a triangle (Diagram 1)
Cut out three copies of one triangle, with its angles marked a, b and c. Copy 2 is Copy 1 slid along line AE, and Copy 3 is Copy 1 turned halfway around, fitted between them. Explain with transversals why the three angles that meet at B form a straight line.
Equation: The half turn makes side BD of Copy 3 parallel to side AC of Copy 1. Line AE is a transversal of the parallel lines AC and BD, so the angle between BE and BD equals angle a (corresponding angles). Line BC is a transversal of the same parallel lines, so the angle between BC and BD equals angle c (alternate interior angles). Together with angle b of Copy 1, the angles at B are b, c and a, and they fill the straight angle ABE. So a + b + c = 180°.
Eight angles from one (Diagram 2, left)
Lines p and q are parallel, and transversal m crosses them. ∠1 measures 118°. Find the other seven angles and give a reason for each.
Equation: ∠2 = 180° - 118° = 62° (a straight line with ∠1). ∠4 = 118° (vertical to ∠1), and ∠3 = 62° (vertical to ∠2). Sliding along m moves each angle onto its corresponding angle, so ∠5 = 118°, ∠6 = 62°, ∠7 = 62° and ∠8 = 118°. Every angle is 118° or 62°.
Alternate interior angles with an unknown
Two parallel lines are cut by a transversal. A pair of alternate interior angles measure (5x - 22)° and (3x + 14)°. Find x and the angle.
Equation: Alternate interior angles are equal: 5x - 22 = 3x + 14, so 2x = 36 and x = 18. Each angle is 5(18) - 22 = 68°. Check: 3(18) + 14 = 68.
An exterior angle (Diagram 2, right)
In triangle PQR, ∠P = 38° and ∠R = 64°. Side PQ is extended past Q. Find the exterior angle x at Q.
Equation: The exterior angle equals the sum of the two remote interior angles: x = 38° + 64° = 102°. Check another way: ∠Q = 180° - 38° - 64° = 78°, and x = 180° - 78° = 102°.
Angle-angle similarity
Triangle JKL has ∠J = 52° and ∠K = 74°. Triangle MNO has ∠M = 52° and ∠O = 54°. Are the triangles similar? Which angles match?
Equation: ∠L = 180° - 52° - 74° = 54°, and ∠N = 180° - 52° - 54° = 74°. Two pairs of angles are equal (∠J = ∠M and ∠L = ∠O), so the triangles are similar, with J matching M, K matching N and L matching O.
Use Diagram 1 with real paper copies: students can see the half turn and the slide. Stress that the tearing in the warm-up shows the fact for one triangle, while the transversal argument works for any triangle, because it only uses facts that are always true.
Guided Practice15 minutes
Pairs solve each problem. One partner finds the numbers and the other says which fact they used; they switch for the next problem.
Guided practice problems with answers
Problem
Answer
Two parallel lines are cut by a transversal. One angle measures 135°. Find its alternate interior angle and its same-side interior angle.
Alternate interior: 135° (equal). Same-side interior: 180° - 135° = 45° (they add up to 180°)
A right triangle has one angle of 29°. Find the third angle.
180° - 90° - 29° = 61°
An exterior angle of a triangle measures 150°. One remote interior angle is 88°. Find the other remote interior angle and the interior angle next to the exterior angle.
Triangle 1 has angles of 40° and 100°. Triangle 2 has two angles of 40°. Are they similar?
Yes. The third angles are 40° and 100°, so both triangles have angles 40°, 40° and 100°
Listen for students who set same-side interior angles equal to each other, and for students who add the exterior angle to the interior angle next to it instead of the two remote ones. Ask them to point to each angle in a sketch before they write an equation.
Independent Practice15 minutes
Students work alone, then compare answers and reasons with a partner.
Independent practice problems with answers
Problem
Answer
Parallel lines are cut by a transversal. Two corresponding angles measure (3x + 8)° and 71°. Find x.
3x + 8 = 71, so x = 21
The angles of a triangle measure (2x)°, (3x)° and (4x)°. Find each angle.
9x = 180, so x = 20: the angles are 40°, 60° and 80°
An exterior angle of a triangle measures 112°, and its two remote interior angles are equal. Find them.
112 ÷ 2 = 56, so each is 56°
One right triangle has a 35° angle. Another right triangle has a 55° angle. Are they similar? Explain.
Yes. The first has angles 90°, 35° and 55°, and so does the second, so two pairs of angles match
Two lines are cut by a transversal, and a pair of corresponding angles measure 80° and 82°. Can the two lines be parallel? Explain.
No. If they were parallel, a slide along the transversal would make the corresponding angles equal
Closure5-10 minutes
Exit ticket: (1) A triangle has angles of 55° and 35°. What is the third angle? (90°, so it is a right triangle.) (2) An exterior angle of a triangle is 140°, and one remote interior angle is 62°. Find the other remote interior angle. (78°.) (3) In one sentence, explain why two pairs of equal angles are enough to make two triangles similar.
Differentiation Strategies
For Struggling Students
Give students a printed copy of Diagram 2 and have them color the four angles of each size in two colors before naming any pairs
Let students keep the torn triangle corners from the warm-up taped in their notes as a reminder of the 180° sum
Use a sentence frame for every answer: "∠___ = ___° because it is ___ to ∠___."
For Advanced Students
Ask students to explain why a triangle can never have two right angles, using the angle sum and then using parallel lines
Have students use the angle sum of a triangle to find the angle sum of a quadrilateral and a pentagon by cutting them into triangles
Extension (beyond this standard): ask whether two pairs of equal angles are enough to show that two quadrilaterals are similar, and find a counterexample with a square and a rectangle
Assessment Guidance
What to Look For
A complete answer names the fact it uses (corresponding, alternate interior, same-side interior, vertical, straight line, angle sum or exterior angle) and does not only give a number. In arguments, look for the reason why a fact holds: the slide along the transversal, the parallel line through a vertex or the three copies of a triangle. Watch for students who treat same-side interior angles as equal, who add all three interior angles to get an exterior angle, and who say two triangles are similar because they both have one right angle.
02
Classroom Activities
3 Activities
1
Three Copies of One Triangle
20 minPairs
This is the official example of the standard. Pairs trace one printed triangle three times onto card stock, cut out the three copies and arrange them as in Diagram 1. They then write an argument with transversals for why the three angles at one point form a straight line.
Procedure
Label the corners of each copy a, b and c, using the same letter on matching corners
Lay Copy 1 on a line drawn across a sheet of paper, then slide Copy 2 along the line until its corner a touches corner b of Copy 1
Turn Copy 3 halfway around and fit it into the gap between them
Lay a ruler through the top corners of Copy 1 and Copy 2 and check that it runs along the top edge of Copy 3, on one line parallel to the first line
Name the transversal that shows each angle at the meeting point is a, b or c, and write the argument in three or four sentences
Discussion Questions
Your triangle and your neighbor's have different shapes. Did the arrangement still make a straight line? Why does the argument not depend on the shape?
How is this argument better than tearing the corners off one triangle, as in the warm-up?
Where in the arrangement do you see a pair of alternate interior angles?
Modification for Distance Learning
Students draw one triangle in free geometry software, make two copies with the translate and rotate tools, and measure the three angles at the meeting point with the angle tool.
2
Slide the Crossing
20 minPairs
Pairs use tracing paper to test the parallel-line facts with a slide and a half turn, then see what changes when the two lines are not parallel.
Procedure
Draw two parallel lines by tracing both edges of a ruler, then draw a slanted transversal across them. Number the eight angles as in Diagram 2
Measure all eight angles with a protractor and record them. Every angle should match one of two measures, and the two measures should add up to 180°
Trace the top crossing on tracing paper. Slide the tracing along the transversal until it covers the bottom crossing, and list the pairs of angles that line up
Now turn the tracing halfway around, with the pencil point at the middle of the transversal segment between the lines, and list the pairs that line up
Repeat the slide with two lines that are not parallel, and describe what goes wrong
Answer Key
The slide matches the corresponding pairs: ∠1 and ∠5, ∠2 and ∠6, ∠3 and ∠7, ∠4 and ∠8. The half turn matches the alternate interior pairs, ∠3 and ∠6, ∠4 and ∠5, and it also swaps the alternate exterior pairs (outside the two lines, on opposite sides of the transversal), ∠1 and ∠8, ∠2 and ∠7. With lines that are not parallel, the tracing of the top line no longer covers the bottom line, and the angle measures at the two crossings are different.
Discussion Questions
Why did the half turn match angles on opposite sides of the transversal, while the slide matched angles on the same side?
If one of the eight angles is a right angle, what are the other seven?
3
Same Two Angles, Different Sizes
20 minGroups of 3
Each student in a group draws a triangle with a 40° angle and a 65° angle at the two ends of a base, but each uses a different base length. The group measures and compares the three triangles to test the angle-angle criterion.
The Three Triangles
Student 1: base 4 cm
Student 2: base 6 cm
Student 3: base 9 cm
Each draws the base, then uses a protractor to draw a 40° angle at the left end and a 65° angle at the right end, and extends the sides until they meet
Procedure
Measure the third angle of each triangle. It should be 75° in all three
Measure the two other sides of each triangle to the nearest 0.1 cm
Divide each side by the base of its own triangle and compare the results across the group
Find the scale factor from the 4 cm triangle to the 6 cm triangle (1.5) and to the 9 cm triangle (2.25), and check that it works for the other sides
Answer Key
Base 4 cm: the side across from the 40° angle is about 2.7 cm, and the side across from the 65° angle is about 3.8 cm
Base 6 cm: about 4.0 cm and 5.6 cm
Base 9 cm: about 6.0 cm and 8.4 cm
Each side divided by its base is about 0.67 and about 0.94 in every triangle, so the triangles are similar
Discussion Questions
You only chose two angles. Why did every group get the same third angle?
Is one pair of equal angles enough? Sketch two triangles that share one angle but have different shapes
Challenge Variation
Each group chooses its own two angles, and each member draws a triangle with a different base. The group swaps its triangles with another group, which must decide which triangles are similar using only a protractor.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Three Copies of a Triangle Make a Straight Line
Three copies of one triangle, drawn to scale. Copy 2 is Copy 1 slid along line AE, and Copy 3 is Copy 1 turned halfway around. Line CD is parallel to line AE, and side BD is parallel to side AC. With lines AE and BC as transversals, the three angles at B are a, b and c, and together they make the straight angle ABE.
Diagram 2: Parallel Lines Cut by a Transversal, and an Exterior Angle
Left: parallel lines p and q are cut by transversal m, making angles 1 to 8, drawn to scale with ∠1 = 118°. Right: triangle PQR with ∠P = 38° and ∠R = 64°; side PQ is extended past Q, and x is the exterior angle at Q.
04
Homework Assignment
~30 min
8.G.A.5 Homework: Angle Facts and Informal Arguments
Directions: Draw a sketch for every problem and label the angles you know. For each angle you find, name the fact you used, and write the explanations in full sentences.
Part 1: Parallel Lines and Transversals (Problems 1-3)
Lines r and s are parallel, and transversal t crosses both. At the crossing with r, the angle above r and to the right of t measures 71°. (a) Find all eight angles. (b) For each angle at the crossing with s, name the fact you used.
Two parallel lines are cut by a transversal. A pair of alternate interior angles measure (4x + 12)° and (6x - 20)°. (a) Find x and the measure of each angle. (b) Explain, using a slide along the transversal and vertical angles, why alternate interior angles are equal.
Two parallel lines are cut by a transversal. A pair of same-side interior angles measure (2x + 5)° and (3x - 15)°. (a) Find x and both angles. (b) Explain why same-side interior angles add up to 180° and are not equal.
Part 2: Triangles and Similarity (Problems 4-6)
The second angle of a triangle is twice the first, and the third angle is 10° more than the second. (a) Find all three angles. (b) Draw a line through one vertex parallel to the opposite side, and use it to explain why the angles add up to 180°.
An exterior angle of a triangle measures 128°, and one of its remote interior angles measures 49°. (a) Find the other remote interior angle. (b) Find the interior angle next to the exterior angle. (c) Explain why the exterior angle equals the sum of the two remote interior angles.
A straight wheelchair ramp starts on level ground and covers 6 m of horizontal distance while it rises 0.5 m. A vertical support post stands under the ramp 2.4 m (measured along the ground) from where the ramp starts. (a) Explain, using angles, why the small triangle under the ramp at the post and the large triangle under the whole ramp are similar. (b) Find the height of the post.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Sketches
Every problem has a labeled sketch
Some sketches missing or unlabeled
No sketches
Named Facts
Every angle comes with the correct fact
Some facts missing or misnamed
No facts named
Arguments
Explanations use slides, parallel lines or the angle sum correctly
Explanations incomplete
No explanations
Accuracy
Every angle and length correct
One or two computing errors
Many errors
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. Question 14 uses Diagram 1. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
Two parallel lines are cut by a transversal. Which pair of angles must have equal measures?
Answer: C
Alternate interior angles are equal: a slide along the transversal and a pair of vertical angles take one onto the other. Choice A is wrong because same-side interior angles add up to 180° and are equal only when both are 90°. Choices B and D describe the same kind of pair, two angles that form a straight line, which also add up to 180°.
Question 2 of 20 · Multiple Choice
Two parallel lines are cut by a transversal. At the top crossing, one angle measures 107°. What does the corresponding angle at the bottom crossing measure?
Answer: B
A slide along the transversal moves the top crossing onto the bottom one, so corresponding angles are equal: 107°. Choice A is 180° - 107°, the angle next to it on the same line. Choice C subtracts from 360° instead of using the slide. Choice D subtracts 90°.
Question 3 of 20 · Multiple Choice
Two parallel lines are cut by a transversal. A pair of alternate interior angles measure (7x - 6)° and (3x + 42)°. What is x?
Answer: D
Alternate interior angles are equal: 7x - 6 = 3x + 42, so 4x = 48 and x = 12. Each angle is 78°. Choice A moves the 6 to the wrong side and gets 4x = 36. Choice B sets the angles' sum equal to 180°, as if they were same-side interior angles. Choice C is the angle measure, not x.
Question 4 of 20 · Multiple Choice
Two parallel lines are cut by a transversal. One same-side interior angle measures 58°. What does the other same-side interior angle measure?
Answer: A
Same-side interior angles add up to 180°, so the other angle is 180° - 58° = 122°. Choice B treats them as equal, which is true only for alternate interior and corresponding angles. Choice C subtracts from 90°. Choice D subtracts from 360°.
Question 5 of 20 · Multiple Choice
Why are corresponding angles equal when a transversal crosses two parallel lines?
Answer: A
Because the lines are parallel, a slide along the transversal carries one line onto the other and each angle onto its corresponding angle, and a slide keeps angle measures. Choice B describes same-side interior angles. Choice C is false: angles next to each other on the transversal are usually different. Choice D confuses corresponding angles, at two crossings, with vertical angles, at one crossing.
Question 6 of 20 · Multiple Choice
Two angles of a triangle measure 36° and 101°. What is the third angle?
Answer: C
The three angles add up to 180°, so the third angle is 180° - 36° - 101° = 43°. Choice A adds the two known angles and stops. Choice B subtracts only 101° from 180°. Choice D subtracts 137° from 360°.
Question 7 of 20 · Multiple Choice
To explain why the angles of triangle ABC add up to 180°, Marco draws a line through C parallel to side AB. Which fact does he use next?
Answer: A
Sides CA and CB are transversals of the parallel lines, so the angles between the new line and those sides equal ∠A and ∠B (alternate interior angles). With ∠C they fill a straight angle at C, so the sum is 180°. Choice B is true but does not connect ∠A and ∠B to C. Choice C is about side lengths, not angles. Choice D is false: an exterior angle and the angle next to it add up to 180°.
Question 8 of 20 · Multiple Choice
One angle of a triangle measures 95°. Which could be the measure of another angle of the same triangle?
Answer: D
95° + 70° = 165°, which leaves 15° for the third angle, so 70° works. Choice A leaves 0° for the third angle, so there would be no triangle. Choices B and C give two angles whose sum is already more than 180°.
Question 9 of 20 · Multiple Choice
In triangle ABC, the exterior angle at C measures 134°, and ∠A = 59°. What is ∠B?
Answer: B
The exterior angle equals the sum of the two remote interior angles: 134° = 59° + ∠B, so ∠B = 75°. Choice A is 180° - 134°, the interior angle at C. Choice C is 180° - 59°. Choice D adds 134° and 59° instead of subtracting.
Question 10 of 20 · Multiple Choice
Why does an exterior angle of a triangle equal the sum of the two remote interior angles?
Answer: A
The exterior angle and its neighbor form a straight line (180°), and the neighbor and the two remote angles also make 180°. Take away the neighbor from both, and the exterior angle equals the two remote angles. Choice B is false: the exterior angle and its neighbor form a straight line, so they add up to 180°. Choice D is false: that total is 180° plus the exterior angle.
Question 11 of 20 · Multiple Choice
Which pair of triangles must be similar?
Answer: B
In choice B, the third angles are 55° and 48°, so both triangles have angles 48°, 55° and 77°. In choice A the angles are 48°, 55°, 77° and 48°, 65°, 67°. Choice C shares only the right angle: 30°, 60°, 90° and 40°, 50°, 90°. Choice D gives 50°, 60°, 70° and 60°, 60°, 60°. One shared angle is not enough.
Question 12 of 20 · Multiple Choice
Why is it enough to check two pairs of angles to show that two triangles are similar?
Answer: A
Once two pairs of angles match, each third angle is 180° minus the same sum, so all three pairs match, and a dilation takes one triangle onto the other. Choice B is false: triangles with the same angles can be any size. Choice C confuses similar with congruent (the same shape and the same size). Choice D is true only for equilateral triangles.
Question 13 of 20 · Multiple Choice
In triangle ABC, point D is on side AB and point E is on side AC, and segment DE is parallel to BC. AD = 4 cm, AB = 10 cm and DE = 3 cm. How long is BC?
Answer: C
∠ADE = ∠ABC (corresponding angles, since DE is parallel to BC), and the triangles share ∠A, so triangle ADE is similar to triangle ABC with scale factor 10 ÷ 4 = 2.5. So BC = 3 × 2.5 = 7.5 cm. Choice A multiplies 3 by 4. Choice B multiplies 3 by 4 ÷ 10, the scale factor upside down. Choice D adds the 6 cm difference between AB and AD to DE.
Question 14 of 20 · Multiple Choice
Maya arranges three copies of a triangle with angles of 35°, 65° and 80° as in Diagram 1. Angle a is 35° and angle b is 65°. What angle does Copy 3 have at point B?
Answer: D
In Diagram 1, Copy 3 has angle c at B, the third angle of the triangle: 180° - 35° - 65° = 80°. Choice A is angle a, which comes from Copy 2. Choice B is angle b, from Copy 1. Choice C is a + b, the angle filled by Copies 1 and 2 together.
Question 15 of 20 · Short Answer
Two parallel lines are cut by a transversal. A pair of corresponding angles measure (2x + 15)° and (3x - 20)°. Find x and the measure of each angle. Then give the measure of an angle at the same crossing that is not equal to them.
Corresponding angles are equal: 2x + 15 = 3x - 20, so x = 35 and each angle is 85°. The angle next to it on the same line is 180° - 85° = 95°.
Question 16 of 20 · Short Answer
The angles of a triangle measure (x + 12)°, (2x + 6)° and (3x - 18)°. Find x and all three angles.
The angles add up to 180°: 6x = 180, so x = 30. The angles are 42°, 66° and 72°. Check: 42 + 66 + 72 = 180.
Question 17 of 20 · Short Answer
An exterior angle of a triangle measures (4x + 6)°. Its two remote interior angles measure 42° and (2x + 4)°. Find x, the exterior angle, and the interior angle next to it.
Exterior angle = sum of the remote interior angles: 4x + 6 = 42 + 2x + 4, so 2x = 40 and x = 20. The exterior angle is 86°, the remote angles are 42° and 44°, and the interior angle next to the exterior angle is 180° - 86° = 94°. Check: 42 + 44 + 94 = 180.
Question 18 of 20 · Short Answer
Triangle RST has ∠R = 35° and ∠S = 80°. Triangle XYZ has ∠X = 65° and ∠Y = 80°. Are the triangles similar? If so, say which angles match.
Yes. ∠T = 180° - 35° - 80° = 65° and ∠Z = 180° - 65° - 80° = 35°. Both triangles have angles 35°, 65° and 80°, so they are similar by angle-angle, with R matching Z, S matching Y and T matching X.
Question 19 of 20 · Short Answer
Explain, using a line parallel to one side, why the three angles of any triangle add up to 180°.
Draw a line through one vertex, C, parallel to the opposite side AB. Side CA is a transversal of the two parallel lines, so the angle between the new line and CA equals ∠A (alternate interior angles). In the same way, the angle on the other side of C equals ∠B. These two angles and ∠C fill the straight line at C, so ∠A + ∠B + ∠C = 180°. The argument works for any triangle, because a parallel line can always be drawn.
Question 20 of 20 · Short Answer
A student who is 1.5 m tall casts a shadow 2 m long. At the same time, a flagpole casts a shadow 12 m long. Explain why the two triangles formed by the objects and their shadows are similar, and find the height of the flagpole.
Both the student and the pole stand straight up, so each triangle has a right angle at the ground. The sun's rays are parallel, so they meet the level ground at equal angles (corresponding angles). Two pairs of equal angles make the triangles similar. The scale factor is 12 ÷ 2 = 6, so the flagpole is 1.5 × 6 = 9 m tall.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does 8.G.A.5 mean?
8.G.A.5 means students explain four angle facts with informal arguments: the angles of a triangle add up to 180°, an exterior angle equals the sum of the two remote interior angles, parallel lines cut by a transversal make pairs of equal angles, and two triangles with two equal angles are similar. The focus is on why the facts are true, not only on using them.
What is an informal argument in 8.G.A.5?
An informal argument is a convincing explanation that uses models and known facts instead of a formal proof. In grade 8, students cut, trace, slide and turn figures, then explain in words why the result must always happen. Formal two-column proofs of the same facts come later, in high school geometry.
What is a transversal in math?
A transversal is a line that crosses two or more other lines. Crossing two lines, it makes eight angles, four at each crossing. When the two lines are parallel, the angles come in pairs: corresponding and alternate interior angles are equal, and same-side interior angles add up to 180°.
Why do the angles of a triangle add up to 180°?
Because they fit together along a straight line. Draw a line through one vertex parallel to the opposite side. The other two angles of the triangle appear again at that vertex as alternate interior angles, and with the third angle they fill a straight angle, 180°. The official example shows the same thing with three copies of one triangle.
What is an exterior angle of a triangle?
It is the angle between one side of a triangle and the extension of the side next to it, outside the triangle. It always equals the sum of the two interior angles that are not next to it, called the remote interior angles. For example, if those angles are 50° and 60°, the exterior angle is 110°.
What is the angle-angle (AA) criterion for similar triangles?
It says that two triangles are similar if two angles of one equal two angles of the other. The third angles then match as well, since each set adds up to 180°. Similar triangles have the same shape, so their matching sides are in one ratio, which lets students find unknown lengths.
Why is AA enough for similar triangles but not for congruent ones?
Equal angles fix the shape of a triangle but not its size. A small triangle and a large one can have the same three angles, so they are similar but not congruent (the same shape and size). Congruent triangles need at least one pair of matching sides as well, which is part of high school geometry.
What mistakes do students make with 8.G.A.5?
A common mistake is setting same-side interior angles equal instead of adding them to 180°. Others use the parallel-line facts for lines that are not parallel, add all three interior angles to get an exterior angle, or decide two triangles are similar because each has one right angle. A labeled sketch catches most of these errors.
How is 8.G.A.5 usually assessed?
Tests usually ask students to find unknown angles in figures with parallel lines or triangles, often with an equation such as 5x - 22 = 3x + 14, and to decide whether two triangles are similar. Many tests also ask for a written explanation, so students should practice naming the fact behind each step.
How does 8.G.A.5 connect to high school geometry?
It is the informal start of several high school proofs. HSG.CO.C.9 proves the facts about lines and angles, HSG.CO.C.10 proves that the angles of a triangle add up to 180°, and HSG.SRT.A.3 proves the angle-angle criterion with similarity transformations. In grade 8, the AA criterion is also used right away in 8.EE.B.6 to explain slope.
07
Related Standards
6 standards
These standards connect to 8.G.A.5: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
7.G.B.5Prerequisite
Use supplementary, complementary, vertical and adjacent angles to find unknown angles
