HSG.SRT.A.3: Proving the AA Similarity Criterion with Transformations
In plain English: HSG.SRT.A.3 is the Common Core geometry standard that asks students to use the properties of similarity transformations to prove the AA criterion: two triangles with two pairs of congruent angles are similar. Students dilate one triangle to the size of the other and then map it with rigid motions. It is usually taught in high school Geometry.
Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
Common Core State Standards for Mathematics · Domain: Similarity, Right Triangles, and Trigonometry (SRT) · Cluster: Understand similarity in terms of similarity transformations Also written as HSG-SRT.A.3 or G-SRT.3 · Official standard
Students already know that two figures are similar when a sequence of rigid motions and dilations maps one onto the other. In this lesson they use two properties of those transformations to prove a shortcut: if two angles of one triangle are congruent to two angles of another triangle, the triangles are similar. Dilations keep angle measures and multiply every length by the same scale factor, and rigid motions keep both lengths and angles.
The proof has two moves. First, dilate one triangle so that one of its sides matches the corresponding side of the other triangle. Second, show that the dilated copy is congruent to the other triangle, so rigid motions finish the mapping. Students then use AA to decide whether triangles are similar, write the correspondence, and find missing lengths in shadow, mirror and crossing-segment situations.
Learning Objectives
By the end of this lesson, students will be able to:
State the properties of dilations and rigid motions that the AA proof depends on: angle measures are preserved, and lengths are multiplied by the scale factor or kept
Prove the AA criterion by dilating one triangle and then mapping the image onto the other triangle with rigid motions
Explain why two pairs of congruent angles are enough for triangles but not for other polygons
Use AA to decide whether two triangles are similar, write the correct correspondence and find a missing side
Describe a specific similarity transformation that maps one triangle onto another on the coordinate plane
Prior Knowledge Required
Students should already be comfortable with:
The definition of similarity using rigid motions and dilations 8.G.A.4
Properties of dilations: angles keep their measure and segments scale by k HSG.SRT.A.1
Deciding whether two figures are similar with similarity transformations HSG.SRT.A.2
The triangle angle sum and vertical angles 8.G.A.5
Give each student a protractor and a ruler. Half the class uses a 5 cm base and half uses an 8 cm base.
Warm-Up Prompt
"Draw a segment for your base. At its left end draw a 40° angle and at its right end draw a 60° angle, and extend the sides until they meet. Measure the other two sides. Then find a partner with the other base length and divide each of your partner's side lengths by the matching side of your triangle. What do you notice?"
Pairs should find three ratios close to 1.6 (8 ÷ 5), with small differences from measuring. Ask: the triangles were built from only two angles and a base, so why do all three sides scale by the same number? Also ask what the third angle measures (80° in every triangle). Tell students that today they prove this is always true, using the transformations they already know.
Direct Instruction20 minutes
Review the two tools. A dilation with scale factor k maps every segment to a segment k times as long and keeps every angle measure. A rigid motion (translation, rotation, reflection) keeps both lengths and angle measures. A similarity transformation is any sequence of these, and two figures are similar when one maps onto the other. Then prove the theorem with Diagram 1.
Given. △ABC and △DEF with ∠A ≅ ∠D and ∠B ≅ ∠E. Goal. Find a similarity transformation that maps △ABC onto △DEF.
Dilate. Let k = DE/AB. Dilate △ABC about A with scale factor k to get △A′B′C′. Then A′B′ = k · AB = DE. Dilations keep angle measures, so ∠A′ ≅ ∠A ≅ ∠D and ∠B′ ≅ ∠B ≅ ∠E.
Compare the image with △DEF. △A′B′C′ and △DEF have A′B′ = DE, ∠A′ ≅ ∠D and ∠B′ ≅ ∠E. By ASA, △A′B′C′ ≅ △DEF, so a sequence of rigid motions maps △A′B′C′ onto △DEF.
Why ASA holds here (the rigid-motion view). Move A′ to D and turn ray A′B′ onto ray DE; because A′B′ = DE, B′ lands on E. Reflect across line DE if needed so C′ is on the same side as F. The angle at D is the same, so ray A′C′ lies on ray DF; the angle at E is the same, so ray B′C′ lies on ray EF. Two lines meet in at most one point, so C′ lands on F.
Conclude. The dilation followed by the rigid motions is a similarity transformation that maps △ABC onto △DEF, so △ABC ~ △DEF. Every side is multiplied by k: DE/AB = EF/BC = DF/AC = k.
Point out where the proof uses the fact that the figures are triangles: the third vertex is pinned down as the intersection of two rays. The angle sum also gives the third pair of angles for free, which is why the criterion is AA and not AAA. Then work the examples below. Before each one, ask students to name the two angle pairs and write the correspondence in the same order.
Finding the second angle pair
In △PQR, m∠P = 48° and m∠Q = 67°. In △STU, m∠S = 48° and m∠U = 65°. Are the triangles similar?
Equation: m∠R = 180° - 48° - 67° = 65° and m∠T = 180° - 48° - 65° = 67°, so ∠P ≅ ∠S and ∠Q ≅ ∠T: △PQR ~ △STU by AA
Naming the similarity transformation
A(0, 0), B(4, 0), C(1, 3) and D(15, 0), E(15, 6), F(10.5, 1.5), with ∠A ≅ ∠D and ∠B ≅ ∠E (Diagram 1).
Equation: k = DE/AB = 6/4 = 3/2. Dilate by 3/2 about A, rotate 90° counterclockwise about A, then translate 15 units right: A → D, B → E, C → F
Shadows (two right angles and the sun's angle)
At the same time of day, a 1.8 m student casts a 2.4 m shadow and a tree casts a 14 m shadow. How tall is the tree?
Equation: Both triangles have a right angle and the same sun angle, so they are similar by AA: h/1.8 = 14/2.4, so h = 10.5 m
Crossing segments (vertical angles)
AE and BD cross at C, ∠A ≅ ∠E, AC = 6, CE = 9 and AB = 8 (Diagram 2). Find ED.
Equation: ∠ACB ≅ ∠ECD (vertical angles), so △ACB ~ △ECD by AA with k = CE/AC = 3/2. ED = 8 · 3/2 = 12
Guided Practice15 minutes
Pairs work on graph paper with △JKL, J(2, 1), K(6, 1), L(2, 4), and △MNP, M(-1, -2), N(-1, -10), P(5, -2). Step 1: show that ∠J and ∠M are right angles and that ∠K ≅ ∠N (for example, by showing the legs of both right triangles are in the ratio 3 : 4). Step 2: find k = MN/JK = 8/4 = 2. Step 3: describe a similarity transformation: dilate by 2 about J, rotate 90° clockwise about J, then translate 3 left and 3 down. Step 4: check that L maps to P. Circulate and ask each pair which step of the AA proof each of their steps matches. Watch for pairs who compute k as JK/MN, which shrinks the triangle instead of enlarging it.
Independent Practice15 minutes
Students work four problems on their own. (1) △ABC has angles of 90° and 27°, and △XYZ has angles of 63° and 90°. Decide whether they are similar. (Yes: the third angle of △ABC is 63°.) (2) A straight ramp rests on the ground. A support post 0.5 m tall stands 2 m from the low end. How tall is a post 5 m from the low end? (1.25 m: the right triangles share the angle at the low end.) (3) △ABC has A(0, 0), B(8, 0), C(0, 6) and △DEF has D(0, 0), E(-4, 0), F(0, -3). Name a similarity transformation. (Dilate by 1/2 about the origin, then rotate 180° about the origin.) (4) Write two sentences explaining why any two right triangles with one pair of congruent acute angles are similar.
Closure5-15 minutes
Exit ticket: (1) Name the two properties of dilations that the AA proof uses. (Angles keep their measure; lengths are multiplied by k.) (2) One triangle has angles of 25° and 105°. Another has angles of 50° and 105°. Are they similar? (Yes: the first triangle's third angle is 50°.) (3) In one sentence, say why the proof would fail for two quadrilaterals. Use the longer time if students present their exit tickets.
Differentiation Strategies
For Struggling Students
Give a proof frame with the five step names (Given, Dilate, Compare, Rigid motions, Conclude) and let students fill in only the reasons
Have students trace the smaller triangle on patty paper, then physically slide, turn or flip it onto the dilated copy
Before writing a similarity statement, have students color-code matching angles so the correspondence order is visible
For Advanced Students
Ask students to prove AA without ASA, using only the rigid-motion argument that places the third vertex at the intersection of two rays
Ask whether AA would still hold on a sphere, where triangle angles add to more than 180°, and what that says about the parallel postulate
Ask students to prove that the ratio of corresponding altitudes of similar triangles equals the scale factor
Assessment Guidance
What to Look For
Listen for students who say that a dilation keeps angle measures and multiplies lengths by k, and who choose k so that one pair of corresponding sides becomes equal. A complete proof names a congruence (ASA or the ray argument) for the second move, not only "the triangles look the same." When students apply AA, check that the similarity statement lists the vertices in corresponding order and that the scale factor is the new length over the original length.
02
Classroom Activities
3 Activities
1
Angle Twins on Patty Paper
20 minPairs
Each partner builds a triangle from the same two angles but a different side length. The pair then carries out the AA proof by hand: a dilation to match one side, then tracing paper to show the rigid motion.
Procedure
The pair chooses two angles that add to less than 180°, such as 35° and 75°
Partner A draws a triangle with those angles on a 6 cm base; Partner B uses a 9 cm base
Partner A dilates the small triangle by 9/6 = 1.5 about one base vertex, measuring along each side from that vertex
The pair traces Partner B's triangle on patty paper and moves it onto the dilated triangle, recording each slide, turn or flip
The pair writes the full similarity transformation and the scale factor
Discussion Questions
Which step used the fact that dilations keep angle measures?
Did anyone need a reflection? How could you tell before moving the paper?
What would change if you started with two angles that add to 180° or more?
Modification for Distance Learning
Use a free dynamic geometry tool: students build both triangles, apply the dilation tool with the ratio of the bases, and drag the image onto the second triangle with the rotate and translate tools.
2
AA Proof Strips
20 minGroups of 3
Groups receive the AA proof cut into 8 strips, each with a statement and a blank for the reason. They put the strips in order and fill in every reason.
The 8 Strips
∠G ≅ ∠L and ∠H ≅ ∠M (given)
Let k = LM/GH (choice of scale factor)
Dilate △GHJ by k to get △G′H′J′, so G′H′ = LM (a dilation multiplies lengths by k)
∠G′ ≅ ∠G and ∠H′ ≅ ∠H (a dilation keeps angle measures)
∠G′ ≅ ∠L and ∠H′ ≅ ∠M (transitive property)
△G′H′J′ ≅ △LMN (ASA)
A sequence of rigid motions maps △G′H′J′ onto △LMN (definition of congruence)
△GHJ ~ △LMN (a dilation followed by rigid motions is a similarity transformation)
Procedure
Hand out the strips shuffled and with the reasons cut off; groups order the statements first
Groups then write the reason for each strip on the back
Two groups compare orders; any strip that could move without breaking the proof is discussed with the class (strips 1 and 2 can swap; strip 4 cannot come before strip 3, which creates △G′H′J′)
Challenge Variation
Remove strip 6 and ask groups to replace it with the ray argument: after the rigid motion sends G′ to L and H′ to M, why must J′ land on N?
3
Similar or Not? Card Sort
25 minGroups of 3-4
Groups sort 8 cards into "similar by AA", "not similar" and "not enough information". For each similar pair they write the correspondence and the scale factor when lengths are given.
The 8 Cards
Angles 42° and 86° in one triangle; 86° and 52° in the other (similar: third angles 52° and 42°)
Two right triangles, one with a 20° angle and one with a 70° angle (similar)
Angles 30° and 60° in one triangle; 30° and 80° in the other (not similar)
Two isosceles triangles, each with a 100° angle (similar: the 100° angle must be the vertex angle, so the base angles are 40°)
Two isosceles triangles, each with a 50° angle (not enough information: 50° could be a base angle or the vertex angle)
Only one pair of congruent angles is given (not enough information)
A triangle and its image after a reflection and a dilation by 3 (similar with k = 3)
A 2-by-5 rectangle and a 3-by-3 square, all angles 90° (not similar: sides 2/3 and 5/3 do not match; AA is only for triangles)
Procedure
Groups sort all cards, then write a one-line reason on each
Each group presents one "not enough information" card and draws two triangles that fit it but are not similar
Close by asking which card shows why the AA proof needs triangles
Discussion Questions
On card 4, why can the 100° angle not be a base angle?
Why is card 8 in the pile, when the AA proof is about triangles?
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: The AA Proof as a Dilation Followed by a Rigid Motion
△ABC has A(0, 0), B(4, 0), C(1, 3) and △DEF has D(15, 0), E(15, 6), F(10.5, 1.5), drawn to scale. The dilation by 3/2 about A makes A′B′ as long as DE and keeps every angle. A 90° rotation about A and a 15-unit translation then carry △A′B′C′ onto △DEF, so △ABC ~ △DEF.
Diagram 2: AA from Vertical Angles
Segments AE and BD cross at C. The given angle pair and the vertical angles at C give two pairs of congruent angles, so △ACB ~ △ECD with scale factor 3/2. Lengths are drawn to scale.
04
Homework Assignment
~30 min
HSG.SRT.A.3 Homework: Proving and Using the AA Criterion
Directions: Show all work. Every similarity statement must list the vertices in corresponding order. In proofs, give a reason for every statement and name the property of dilations or rigid motions you use.
Part 1: Proving AA (Problems 1-2)
△ABC and △XYZ have ∠A ≅ ∠X and ∠C ≅ ∠Z. Write a proof that △ABC ~ △XYZ. Use a dilation with scale factor k = XZ/AC, then explain why the image is congruent to △XYZ. Name the property of dilations used at each step.
△RST has R(1, 1), S(5, 1), T(1, 4). △UVW has U(-2, 2), V(-10, 2), W(-2, 8). (a) Find UV/RS, UW/RT and VW/ST. (b) Describe a similarity transformation that maps △RST onto △UVW. (c) Explain how your transformation shows that ∠R ≅ ∠U and ∠S ≅ ∠V.
Part 2: Deciding Similarity (Problems 3-4)
Decide whether each pair of triangles must be similar. Explain, and write a similarity statement when they are. (a) △ABC with m∠A = 52°, m∠B = 71°, and △DEF with m∠E = 71°, m∠F = 57°. (b) △GHI with m∠G = 35°, m∠H = 90°, and △JKL with m∠J = 35°, m∠K = 65°. (c) Two isosceles triangles whose vertex angles both measure 40°.
A student whose eyes are 1.6 m above the ground puts a small mirror flat on the ground 2 m in front of her feet. Standing still, she sees the top of a flagpole in the mirror. The mirror is 7.5 m from the base of the flagpole. The angle of incidence equals the angle of reflection. Explain why two triangles in this situation are similar, then find the height of the flagpole.
Part 3: Algebra and Limits of AA (Problems 5-6)
In △ABC and △DEF, m∠A = m∠D = 50°, m∠B = (2x + 10)° and m∠E = (3x - 15)°. (a) Find the value of x that makes △ABC ~ △DEF with B corresponding to E. (b) Find m∠C and m∠F. (c) If AB = 6, DE = 9 and BC = 10, find EF.
Rectangle ABCD is 4 cm by 6 cm. (a) Give the dimensions of a rectangle that is similar to ABCD with scale factor 3/2, and of a rectangle with the same four angles that is not similar to ABCD. Show the side ratios. (b) Identify the step of the AA proof that fails for quadrilaterals and explain why.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Proof Structure
Dilation, congruence and conclusion in order, each with a reason
Correct steps with a missing reason
Steps missing or out of order
Transformation Properties
Names angle preservation and length scaling where used
Names one property
No properties named
Correspondence and Scale Factor
Vertices in matching order, k computed as new over original
Minor order or ratio error
Correspondence missing or wrong
Applications
Correct lengths and angles with the AA justification
Correct setup with an arithmetic error
Incorrect or unjustified
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. 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
In the proof of the AA criterion, why does the dilated triangle △A′B′C′ have the same angle measures as △ABC?
Answer: C
A dilation multiplies lengths by k but keeps every angle measure, so ∠A′ ≅ ∠A and ∠B′ ≅ ∠B. Choice A is a property of rigid motions, not of dilations with k ≠ 1. Choice B confuses angles with lengths.
Question 2 of 20 · Multiple Choice
In △ABC, m∠A = 38° and m∠B = 74°. In △DEF, m∠D = 38° and m∠F = 68°. Which statement is true?
Answer: B
m∠C = 180° - 38° - 74° = 68° and m∠E = 180° - 38° - 68° = 74°. So A ↔ D, B ↔ E and C ↔ F, and △ABC ~ △DEF. Choice A lists the vertices in the wrong order. Choice C forgets that the angle sum gives the missing angles. Choice D claims congruence, but no side lengths are known.
Question 3 of 20 · Multiple Choice
To prove △ABC ~ △DEF by AA, a student dilates △ABC with scale factor k. Which value of k makes the proof work with ∠A ≅ ∠D and ∠B ≅ ∠E?
Answer: B
The dilation must make A′B′ = DE, and A′B′ = k · AB, so k = DE/AB. Choice A is the reciprocal, which makes A′B′ = AB²/DE. Choices C and D are not scale factors of a dilation.
Question 4 of 20 · Multiple Choice
After the dilation in the AA proof, △A′B′C′ and △DEF have A′B′ = DE, ∠A′ ≅ ∠D and ∠B′ ≅ ∠E. Which criterion shows they are congruent?
Answer: D
The congruent side A′B′ = DE lies between the two pairs of congruent angles, which is ASA. Choice A is not a valid congruence criterion. Choice C only shows similarity, and choice B needs all three sides.
Question 5 of 20 · Multiple Choice
Which pair of triangles must be similar?
Answer: A
Each right triangle has a 90° angle and a 35° angle, which gives two pairs of congruent angles, so AA applies. Choice B gives only one angle pair. Choice C fails because isosceles triangles can have any vertex angle.
Question 6 of 20 · Multiple Choice
At the same time of day, a 2 m pole casts a 3.2 m shadow and a building casts a 24 m shadow. How tall is the building?
Answer: C
Both triangles have a right angle and the same sun angle, so they are similar by AA: h/2 = 24/3.2, so h = 15 m. Choice A inverts the ratio (3.2 · 24/2). Choice B subtracts the difference of the pole's measurements from 24.
Question 7 of 20 · Multiple Choice
Segments PS and QR cross at M. ∠Q ≅ ∠R, PM = 4, QM = 6, RM = 9 and PQ = 5. P corresponds to S. Find SR.
Answer: B
∠PMQ ≅ ∠SMR (vertical angles) and ∠Q ≅ ∠R, so △PMQ ~ △SMR by AA with k = RM/QM = 9/6 = 3/2. SR = 5 · 3/2 = 7.5. Choice A uses the reciprocal scale factor. Choice C is SM, the side that matches PM, not SR.
Question 8 of 20 · Multiple Choice
△ABC has m∠A = 40° and m∠B = 75°. △DEF has m∠D = 40° and m∠E = 75°. A student says she must also measure ∠C and ∠F before she can use AA. What is the best response?
Answer: C
m∠C = 180° - 40° - 75° = 65° and m∠F = 180° - 40° - 75° = 65°, so the third pair matches automatically. That is why the criterion needs only two angles. Choice A treats the third pair as new information, but the angle sum already fixes it. Choice D confuses similarity with congruence.
Question 9 of 20 · Multiple Choice
△ABC has A(0, 0), B(2, 0), C(0, 3) and △DEF has D(1, 1), E(7, 1), F(1, 10). Which similarity transformation maps △ABC onto △DEF?
Answer: A
Dilating by 3 about the origin gives (0, 0), (6, 0), (0, 9), and moving 1 right and 1 up gives D(1, 1), E(7, 1), F(1, 10). Choice B translates first, so the dilation sends A to (3, 3), not D. Choice D leaves A at the origin.
Question 10 of 20 · Multiple Choice
Which pair of figures has all corresponding angles congruent but is NOT similar?
Answer: D
All angles are 90°, but the ratios of the sides are 4/3 and 4/4 = 1, which are not equal, so no dilation matches them. This is why AA is a triangle criterion only. Choices A, B and C are similar pairs.
Question 11 of 20 · Multiple Choice
△JKL has a right angle at K and m∠J = 31°. △MNP has a right angle at N and m∠P = 59°. Which similarity statement is correct?
Answer: B
m∠L = 180° - 90° - 31° = 59° = m∠P and m∠M = 180° - 90° - 59° = 31° = m∠J. So J ↔ M, K ↔ N, L ↔ P. Choice A pairs J with P, but ∠J is 31° and ∠P is 59°. Choice D ignores the third angles.
Question 12 of 20 · Multiple Choice
In the rigid-motion step of the AA proof, A′ is moved to D and B′ to E, with C′ on the same side of line DE as F. Why must C′ land on F?
Answer: A
Because the angles at D and E match, ray A′C′ lies on ray DF and ray B′C′ lies on ray EF. Two distinct lines meet in at most one point, so C′ = F. Choice C names the vertices without giving a reason.
Question 13 of 20 · Multiple Choice
In △ABC and △DEF, m∠B = m∠E = 72°, m∠A = (5x - 7)° and m∠D = (3x + 15)°. For which value of x is △ABC ~ △DEF?
Answer: C
Set 5x - 7 = 3x + 15, so 2x = 22 and x = 11. Then both angles measure 48°, and with the 72° angles that is AA. Choice B stops at 2x = 22. Choice D is the angle measure, not x.
Question 14 of 20 · Multiple Choice
One isosceles triangle has a base angle of 55°. Another isosceles triangle has a vertex angle of 70°. Are they similar?
Answer: B
The first triangle's vertex angle is 180° - 2(55°) = 70°. The second triangle's base angles are (180° - 70°)/2 = 55°. The angles match, so AA applies. Choice A stops before finding the missing angles.
Question 15 of 20 · Short Answer
Write the outline of a proof that △PQR ~ △STU when ∠Q ≅ ∠T and ∠R ≅ ∠U. Name the scale factor and the property used in each step.
Dilate △PQR by k = TU/QR to get △P′Q′R′ with Q′R′ = TU (dilations multiply lengths by k). Angles: ∠Q′ ≅ ∠Q ≅ ∠T and ∠R′ ≅ ∠R ≅ ∠U (dilations keep angle measures). Congruence: △P′Q′R′ ≅ △STU by ASA, because the side Q′R′ = TU lies between the two angle pairs, so rigid motions map one onto the other. Conclusion: the dilation followed by those rigid motions maps △PQR onto △STU, so △PQR ~ △STU.
Question 16 of 20 · Short Answer
A person 1.7 m tall stands 6 m from a lamppost. Her shadow is 2 m long. How tall is the lamppost? Justify the similarity.
The person and the lamppost both make right angles with the ground, and the two triangles share the angle at the tip of the shadow, so they are similar by AA. The large triangle's base is 6 + 2 = 8 m. h/1.7 = 8/2, so h = 6.8 m. A common error is to use 6 m instead of 8 m for the large base.
Question 17 of 20 · Short Answer
△RST has R(0, 0), S(4, 0), T(0, 2). △XYZ has X(0, 0), Y(0, -6), Z(3, 0). Find the scale factor and describe a similarity transformation that maps △RST onto △XYZ.
XY/RS = 6/4 and XZ/RT = 3/2, so k = 3/2. Dilate by 3/2 about the origin to get (0, 0), (6, 0), (0, 3). Then rotate 90° clockwise about the origin, (x, y) → (y, -x): (6, 0) → (0, -6) = Y and (0, 3) → (3, 0) = Z. Both steps keep angle measures, so ∠R ≅ ∠X and ∠S ≅ ∠Y.
Question 18 of 20 · Short Answer
Explain with a counterexample why one pair of congruent angles is not enough to prove two triangles similar.
A triangle with angles 30°, 50° and 100° and a triangle with angles 30°, 75° and 75° share a 30° angle. Their other angles do not match, and a similarity transformation keeps every angle, so no similarity transformation can map one onto the other. They are not similar. A second pair of congruent angles is needed.
Question 19 of 20 · Short Answer
Use AA to explain why any two equilateral triangles are similar. Then give the scale factor from an equilateral triangle with side 4 to one with side 10.
Every angle of an equilateral triangle measures 60°, so any two equilateral triangles have two pairs of congruent angles and are similar by AA. The scale factor is 10/4 = 5/2.
Question 20 of 20 · Short Answer
In right triangle ABC, the right angle is at C. Point E is on AC and point D is on AB, with DE ⊥ AC. AC = 12, BC = 9 and AE = 8. Find DE and name the similar triangles.
∠AED and ∠ACB are right angles and ∠A is shared, so △AED ~ △ACB by AA with k = AE/AC = 8/12 = 2/3. DE = (2/3) · 9 = 6.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does HSG.SRT.A.3 mean?
HSG.SRT.A.3 means students prove the AA criterion from the properties of similarity transformations, instead of taking it as a rule. The proof uses a dilation, which keeps angles and scales lengths, followed by rigid motions, which keep both. After the proof, AA becomes the main tool for showing triangles are similar.
Is HSG.SRT.A.3 taught in Geometry or in Algebra?
It is a high school Geometry standard. It sits in the first cluster of the Similarity, Right Triangles, and Trigonometry domain, right after students define similarity with transformations (HSG.SRT.A.1 and HSG.SRT.A.2) and before they prove theorems with similar triangles (HSG.SRT.B.4).
Why does AA need only two angles and not three?
Because the angles of a triangle add to 180°, the third angle is set once two are known. If two angles match, the third angles are 180° minus the same total, so they match as well. Students can call it AAA, but the third condition adds nothing.
Why prove AA with transformations instead of just using it?
In Common Core Geometry, "similar" is defined by transformations: one figure maps onto the other by rigid motions and dilations. AA is not part of that definition, so it has to be proved from it. The proof also shows students why the sides end up proportional: the dilation multiplies every side by the same k.
Which properties of dilations does the AA proof use?
Two properties: a dilation keeps every angle measure, and it multiplies every length by the scale factor k. The proof chooses k so that one side of the image equals the corresponding side of the other triangle. It then uses the rigid-motion properties from HSG.CO.B.8, which keep lengths and angles, to finish the mapping.
Does AA work for quadrilaterals or other polygons?
No. A 1-by-4 rectangle and a 2-by-2 square have all angles equal to 90° but are not similar. In the proof, the third vertex of a triangle is fixed as the point where two rays meet. A quadrilateral has a fourth vertex that angles alone do not fix, so side ratios can differ.
What mistakes do students make with the AA criterion?
A frequent one is writing the similarity statement in the wrong order, for example △ABC ~ △DFE when B matches E. Others are using a single pair of angles, forgetting to find the third angle with the angle sum, and computing the scale factor upside down. Ask students to mark matching angles before writing anything.
How is the AA criterion usually assessed?
Expect three kinds of items: a proof or proof outline that uses a dilation and rigid motions, a decision about whether two triangles are similar given some angles, and an applied problem, such as a shadow or mirror situation, where students justify AA and then find a length. Coordinate items may ask for a specific similarity transformation.
Can I use the SAS or SSS similarity criteria in this lesson?
This standard is about AA only. SAS and SSS similarity are used later, together with congruence criteria, when students solve problems and prove relationships (HSG.SRT.B.5). Keeping this lesson focused on AA lets students see the full proof without mixing criteria.
How does HSG.SRT.A.3 lead to trigonometry?
Any two right triangles with the same acute angle are similar by AA, so their side ratios are equal. That is why sine, cosine and tangent depend only on the angle (HSG.SRT.C.6). Without AA, the trigonometric ratios would not be well defined.
07
Related Standards
6 standards
These standards connect to HSG.SRT.A.3: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
8.G.A.4Prerequisite
Similar figures: one maps to the other by rigid motions and dilations
