HSG.SRT.A.2: Deciding Similarity with Similarity Transformations
In plain English: HSG.SRT.A.2 is the Common Core geometry standard that asks students to decide whether two figures are similar by looking for a sequence of rigid motions and dilations that maps one onto the other. For triangles, students explain with these transformations why similarity means all corresponding angles are equal and all corresponding sides are proportional. It is usually taught in high school Geometry.
Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
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.2 or G-SRT.2 · Official standard
Students learn the transformation definition of similarity: two figures are similar when a sequence of rotations, reflections, translations and dilations maps one onto the other. They use it both ways. To show that two figures are similar, they find and verify such a sequence. To show that two figures are not similar, they explain why no sequence can work, usually because no single scale factor fits every pair of sides or because an angle would have to change.
For triangles, students then explain what the definition means in terms of measurements: rigid motions and dilations keep angle measures and multiply every length by the same factor, so similar triangles have all three pairs of corresponding angles equal and all three pairs of corresponding sides proportional. Students also explain the reverse direction, which sets up the AA criterion in HSG.SRT.A.3.
Learning Objectives
By the end of this lesson, students will be able to:
State the definition of similarity in terms of similarity transformations
Find and verify a sequence of rigid motions and dilations that maps one figure onto another, with coordinates or tracing paper
Explain why two figures are not similar by showing that no single scale factor or angle-preserving sequence can map one onto the other
Explain, using the properties of rigid motions and dilations, why similar triangles have equal corresponding angles and proportional corresponding sides
Use a scale factor and corresponding parts to find unknown sides and angles of similar triangles
Prior Knowledge Required
Students should already be comfortable with:
Similar figures described as the result of rotations, reflections, translations and dilations 8.G.A.4
Properties of dilations: parallel images and lengths multiplied by the scale factor HSG.SRT.A.1
Congruence defined by rigid motions, which preserve distance and angle measure HSG.CO.B.6
Coordinate rules for translations, reflections across the axes and rotations of 90° and 180° about the origin
Show three rectangles: an original phone photo 4 in by 3 in, a copy resized to 8 in by 6 in, and a copy stretched to 8 in by 3 in. Post the prompt:
Warm-Up Prompt
"Which copy looks like the original, only bigger? Describe exactly what was done to each side of the photo in each copy. What would you have to do to the original to get the stretched copy, and is that a dilation?"
Students should notice that the 8 by 6 copy multiplies both sides by 2, while the stretched copy multiplies the width by 2 and leaves the height alone. Connect this to HSG.SRT.A.1: a dilation multiplies every length by the same factor, so the stretched copy cannot come from any dilation, and it will not be similar to the original. Tell students that today they will turn "looks the same, only bigger" into a precise test.
Direct Instruction20 minutes
Definition. A similarity transformation is a rigid motion, a dilation, or any sequence of them. Two figures are similar if a similarity transformation maps one onto the other. To decide whether figures are similar, students either find the sequence and check that it lands exactly on the second figure, or explain why none can exist.
Why angles stay equal and sides stay proportional. Rigid motions keep every length and every angle measure. A dilation with scale factor k keeps every angle measure (it maps each side of an angle to a parallel side) and multiplies every length by k. So after any sequence, every angle keeps its measure and every length is multiplied by the product of the scale factors. Diagram 1 shows a two-step sequence and Diagram 2 shows the resulting measurements for a triangle.
Similar: dilation, then translation
△ABC has A(0, 0), B(2, 0), C(0, 1), and △DEF has D(1, 1), E(7, 1), F(1, 4).
Equation: Dilate by 3 about the origin: (0, 0), (6, 0), (0, 3). Translate by (1, 1): D, E, F. So △ABC ~ △DEF
Not similar: no single scale factor
A rectangle is 4 cm by 6 cm and another is 6 cm by 10 cm.
Equation: 6/4 = 1.5 but 10/6 ≈ 1.67, so no dilation fits both sides: not similar
Similar: rotation, then dilation
△PQR has P(0, 0), Q(4, 0), R(0, 2), and △STU has S(0, 0), T(0, -2), U(1, 0).
Equation: Rotate 90° clockwise about the origin, then dilate by 1/2: P → S, Q → T, R → U
Using the meaning for triangles
△ABC ~ △DEF with AB = 4, BC = 6, AC = 5 and DE = 6.
Equation: k = 6/4 = 1.5, so EF = 9 and DF = 7.5, and ∠D = ∠A, ∠E = ∠B, ∠F = ∠C
The reverse direction
In △ABC and △DEF, all three pairs of angles are equal and DE/AB = EF/BC = DF/AC = k.
Equation: Dilate △ABC by k: its sides become DE, EF, DF. A rigid motion then maps it onto △DEF, so △ABC ~ △DEF
For the last example, spell out why the rigid motion exists: after the dilation the new triangle has the same three side lengths as △DEF, so it is congruent to △DEF (HSG.CO.B.8), and congruent figures are related by a rigid motion. This is why, for triangles, "similar" means exactly "all corresponding angles equal and all corresponding sides proportional". Point out that for other polygons one of these conditions alone is not enough: a square and a long rectangle both have four right angles.
Guided Practice15 minutes
Pairs work on grid paper with tracing paper available. For each pair of figures, they either write a sequence and verify every vertex, or explain why no sequence exists.
Triangle (0, 0), (3, 0), (0, 4) and triangle (2, 1), (8, 1), (2, 9). (Dilate by 2 about the origin, then translate by (2, 1): similar.)
Triangle (0, 0), (3, 0), (0, 4) and triangle (0, 0), (6, 0), (0, 5). (6/3 = 2 but 5/4 = 1.25: not similar.)
Any two squares, with sides s and t. (Dilate the first by t/s, then use a rigid motion to place it on the second: always similar.)
After each item, ask one pair to explain the order of their sequence. Point out that "translate by (2, 1), then dilate by 2 about the origin" is a different sequence with a different result, because the dilation also doubles the translation. Listen for students who compare only one pair of sides and stop.
Independent Practice15 minutes
Triangle (1, 1), (3, 1), (1, 2) and triangle (3, -3), (9, -3), (3, -6). Find a sequence of transformations that maps the first onto the second. (Reflect across the x-axis, then dilate by 3 about the origin.)
A rectangle measures 3 m by 5 m and another measures 6 m by 9 m. Decide whether they are similar and justify. (6/3 = 2 but 9/5 = 1.8: not similar.)
A triangle has sides 5, 12 and 13, and another has sides 7.5, 18 and 19.5. Find the scale factor, and find the angle opposite the side 19.5 if the angle opposite 13 is a right angle. (k = 1.5; 90°.)
Write two or three sentences explaining why a dilation cannot change the measure of an angle.
Closure5-10 minutes
Exit ticket: (1) Complete the sentence: "Two figures are similar if..." (2) △JKL ~ △MNP with JK = 8 and MN = 2. What is the scale factor from △JKL to △MNP, and how does ∠K compare with ∠N? (1/4; they are equal.) (3) Name one reason two rectangles might fail to be similar.
Differentiation Strategies
For Struggling Students
Give a checklist for each pair of figures: match the vertices, compute every ratio of corresponding sides, compare the angles, then name the transformations
Let students do the rigid motion with tracing paper first and the dilation with coordinates second, so they handle one kind of transformation at a time
Start with pairs where only a dilation is needed, then add one rigid motion at a time
For Advanced Students
Ask for two different sequences that map the same triangle onto the same image, and explain why both are valid
Ask students to prove that the composition of a dilation with factor 2 and a dilation with factor 3, with different centers, multiplies every length by 6
Ask why any two circles are similar (HSG.C.A.1) and describe the sequence for two specific circles
Assessment Guidance
What to Look For
Check that students verify a proposed sequence on every vertex, not only on one, and that they state the transformations in order with their centers, lines or vectors. When figures are not similar, look for a specific reason, such as two different side ratios, rather than "they look different". For the triangle explanation, listen for the two properties that carry the argument: rigid motions and dilations preserve angle measure, and every length is multiplied by the same scale factor.
02
Classroom Activities
3 Activities
1
Transformation Detective
20 minPairs
Pairs receive 6 cards, each showing two figures on a coordinate grid. For each card they either write a similarity transformation that maps the first figure onto the second and verify it vertex by vertex, or explain why none exists.
Card 1: dilate by 2 about the origin, translate by (3, 3). Card 2: rotate 90° counterclockwise about the origin; the figures are congruent, which is similarity with k = 1
Card 3: 4/2 = 2 but 5/3 ≈ 1.67, not similar. Card 4: reflect across the y-axis, dilate by 2 about the origin
Card 5: dilate by 3 about the origin, translate by (1, 0). Card 6: 6/3 = 2 but 5/3 ≈ 1.67, not similar
Modification for Distance Learning
Post the cards in geometry software. Students apply the transformation tools to the first figure and share a screenshot showing the image landing on the second figure, or a note explaining why it cannot.
2
Measure the Evidence: Angles Stay, Sides Scale
20 minGroups of 3
Each group builds a similar triangle through a sequence of transformations by hand and then measures it, so that the explanation for triangles rests on their own data.
Procedure
Student 1 draws a scalene triangle with sides between 3 cm and 7 cm and dilates it by k = 2 from a center outside it, using a ruler along the rays
Student 2 traces the dilated triangle on tracing paper, then rotates or flips the tracing paper and copies the triangle onto a new spot on the page
Student 3 measures all three sides and all three angles of the original and the final triangle and records them in a two-column table, plus a column of side ratios
The group writes one paragraph: which step changed the lengths, which steps changed nothing, and why no step changed an angle
Discussion Questions
Your three side ratios were close to 2. Could one of them honestly have come out 3?
If the final triangle has the same angles as the original, does that prove the sides are proportional? What would you check?
How would the table change if you had dilated by k = 1/2 instead?
3
Similar or Not? Card Sort
15-20 minGroups of 3-4
Groups sort 8 cards into "always similar", "similar for these measurements" and "not similar", and write a one-sentence justification that names a transformation or a failed ratio for each card.
The 8 Cards
Two circles of radius 2 cm and 5 cm
Two squares with sides 3 in and 7 in
A 4 by 6 rectangle and a 6 by 9 rectangle
Two isosceles triangles with vertex angles 40° and 50°
A right triangle with legs 3 and 4 and a right triangle with legs 6 and 8
Two rhombuses with sides 4 and 6 that each have a 70° angle
Two regular hexagons with sides 1 cm and 2.5 cm
A triangle with sides 2, 3, 4 and a triangle with sides 4, 6, 7
Challenge Variation
Groups create one extra card that tricks the class: two figures with all angles equal that are not similar, or two figures with all sides proportional that are not similar. They must explain why their card works.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: A Similarity Transformation in Two Steps
Triangle PQR with P(0, 0), Q(4, 0), R(0, 2), drawn to scale. A 90° clockwise rotation about the origin gives the dashed triangle, and a dilation with center (0, 0) and scale factor 1/2 then gives △STU. Because a sequence of rigid motions and dilations maps △PQR onto △STU, the triangles are similar.
Diagram 2: What Similarity Means for Triangles
A triangle with sides 4, 5 and 6 and its image after a dilation with scale factor 1.5 and a translation, drawn to scale. Each angle keeps its measure, and each side is multiplied by 1.5, so all corresponding angles are equal and all corresponding sides are proportional.
04
Homework Assignment
~30 min
HSG.SRT.A.2 Homework: Similarity Transformations
Directions: Show all work. When you claim two figures are similar, name every transformation in order (with its center, line or vector and scale factor) and check every vertex or side. When you claim they are not similar, give a specific reason.
Part 1: Deciding Whether Figures Are Similar (Problems 1-3)
△GHI has G(0, 0), H(5, 0), I(0, 2), and △G′H′I′ has G′(0, 0), H′(0, 10), I′(-4, 0). Find a sequence of a rotation and a dilation that maps △GHI onto △G′H′I′, and verify it for all three vertices. What is the scale factor?
Rectangle A is 9 cm by 12 cm, rectangle B is 6 cm by 8 cm, and rectangle C is 6 cm by 9 cm. Which rectangles are similar to rectangle A? For each one that is, give the scale factor from A. For each one that is not, explain why no similarity transformation can map A onto it.
Rhombus WXYZ has sides of 5 cm and a 60° angle. (a) Is it similar to a square with sides of 5 cm? Explain using transformations. (b) Is it similar to a rhombus with sides of 8 cm and a 120° angle? Explain.
Part 2: What Similarity Means for Triangles (Problems 4-6)
△ABC ~ △KLM with A ↔ K, B ↔ L and C ↔ M. AB = 14, BC = 10, AC = 8 and LM = 25. (a) Find the scale factor from △ABC to △KLM. (b) Find KL and KM. (c) m∠C ≈ 101.5°. Find m∠M and explain why it must have that value.
Suppose a sequence of rigid motions and dilations maps △RST onto △XYZ, with R → X, S → Y and T → Z. Explain, step by step, why ∠S ≅ ∠Y and why RS/XY = ST/YZ = RT/XZ.
A school logo is a triangle with sides 3 cm, 4 cm and 6 cm. (a) A banner version must have its longest side 90 cm. Find the other two sides of a similar banner triangle. (b) A student's banner triangle has sides 48 cm, 60 cm and 90 cm. Is it similar to the logo? Explain.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Sequence of Transformations
Every transformation named in order and verified on all vertices
Sequence correct but not fully verified
No sequence or an incorrect one
Not-Similar Reasoning
Specific failed ratio or changed angle named
Correct conclusion with a vague reason
Incorrect conclusion
Corresponding Parts
Scale factor, sides and angles all correct
One error in a side or angle
Several errors
Explanation for Triangles
Uses angle and length properties of each transformation
Uses the properties but skips a step
No transformation-based explanation
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Choose an answer to see whether it is right and why. Work the short-answer questions on grid paper before you open the answer. Reset quiz clears all answers.
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
Which statement is the definition of similar figures used in this standard?
Answer: B
The standard defines similarity through similarity transformations: rigid motions and dilations. Choice A is the informal idea the definition makes precise. Choice C is not enough for figures other than triangles: a square and a long rectangle have equal angles but are not similar.
Question 2 of 20 · Multiple Choice
△ABC has A(0, 0), B(2, 0), C(0, 3). Which sequence maps it onto the triangle with vertices (1, 2), (5, 2), (1, 8)?
Answer: C
Dilating by 2 gives (0, 0), (4, 0), (0, 6); translating by (1, 2) gives (1, 2), (5, 2), (1, 8). Choice A does the steps in the other order and lands on (2, 4), (6, 4), (2, 10), because the dilation also doubles the translation. Choice B uses the wrong scale factor: 5 - 1 = 4 is twice AB = 2.
Question 3 of 20 · Multiple Choice
A rectangle is 5 cm by 8 cm. Another is 10 cm by 14 cm. Are they similar?
Answer: D
A dilation multiplies every length by the same factor, and no rigid motion changes lengths, so the ratios of corresponding sides must all be equal. They are 2 and 1.75, so no similarity transformation works. Choice B checks only one pair of sides. Choice C gives a wrong reason: similar figures usually have different areas.
Question 4 of 20 · Multiple Choice
△ABC ~ △DEF with A ↔ D, B ↔ E and C ↔ F. If AB = 6, DE = 9 and BC = 8, what is EF?
Answer: A
The scale factor is 9/6 = 1.5, so EF = 1.5 · 8 = 12. Choice B adds 3 to BC, the same amount that was added to AB, but similarity multiplies lengths. Choice C uses the ratio upside down: 8 · 6/9.
Question 5 of 20 · Multiple Choice
△ABC ~ △XYZ with A ↔ X, B ↔ Y and C ↔ Z, and the scale factor is 1.5. If m∠A = 48° and m∠B = 71°, what is m∠Z?
Answer: D
m∠C = 180° - 48° - 71° = 61°, and corresponding angles of similar triangles are equal, so m∠Z = 61°. Choice B multiplies 61° by the scale factor, but dilations do not change angle measures. Choices A and C are the measures of the other two angles.
Question 6 of 20 · Multiple Choice
Why do corresponding angles of similar triangles have equal measures?
Answer: A
Similar triangles are related by a sequence of rigid motions and dilations, and each of these keeps every angle measure. Choice B is false: only lengths are multiplied. Choice C is true for all triangles but does not explain why the individual angles match. Choice D is true only for a single dilation, not after a rotation.
Question 7 of 20 · Multiple Choice
Which pair of figures is always similar?
Answer: D
Any two regular pentagons have all angles equal to 108° and all sides in the same ratio, so a dilation by the ratio of their sides followed by a rigid motion maps one onto the other. Rectangles can have different length-to-width ratios (choice A), and isosceles triangles and rhombuses can have different angles (choices B and C).
Question 8 of 20 · Multiple Choice
One triangle has sides 3, 5 and 7. Another has sides 6, 10 and 12. What can you conclude?
Answer: B
For similarity every pair of corresponding sides needs the same ratio: 6/3 = 2 and 10/5 = 2, but 12/7 ≈ 1.71. Choice C stops after two pairs, a common error. Choice D gives a wrong reason, since size does not matter.
Question 9 of 20 · Multiple Choice
△PQR has P(0, 0), Q(3, 0), R(0, 3). It is dilated by 1/3 about the origin and then reflected across the y-axis. What is the image of Q?
Answer: B
The dilation sends Q(3, 0) to (1, 0), and the reflection across the y-axis changes the sign of x: (-1, 0). Choice A skips the reflection. Choice C skips the dilation. Choice D reflects across the line y = -x instead of the y-axis.
Question 10 of 20 · Multiple Choice
A student says a triangle with sides 4, 6, 8 is similar to a triangle with sides 6, 8, 10 because "each side grew by 2". What is the error?
Answer: C
A dilation multiplies every length by the same factor, so the ratios must match. Adding the same amount to each side changes the shape. Choice D is not the error: comparing sides is a valid test, as long as ratios are compared.
Question 11 of 20 · Multiple Choice
△ABC ~ △DEF with scale factor 3 from △ABC to △DEF. The perimeter of △ABC is 14 cm. What is the perimeter of △DEF?
Answer: A
Each side of △DEF is 3 times the matching side of △ABC, so the sum is 3 times as large: 3 · 14 = 42 cm. Choice B adds 3 instead of multiplying. Choice C multiplies by 3² = 9, which is how area would change, not perimeter.
Question 12 of 20 · Multiple Choice
△ABC ~ △DEF with A ↔ D and B ↔ E. If AB = 10 and DE = 4, what is the scale factor from △ABC to △DEF?
Answer: B
The scale factor from the first triangle to the second is DE/AB = 4/10 = 0.4, a reduction. Choice A is the scale factor from △DEF back to △ABC. Choice C is the difference of the sides.
Question 13 of 20 · Multiple Choice
The point (3, -1) is rotated 180° about the origin and then dilated by 2 about the origin. What is its final image?
Answer: C
The rotation gives (-3, 1), and the dilation doubles both coordinates: (-6, 2). Choice A skips the rotation. Choice B skips the dilation. Choice D adds 2 to each coordinate of (-3, 1) instead of multiplying.
Question 14 of 20 · Multiple Choice
△ABC ~ △DEF with A ↔ D, B ↔ E, C ↔ F and scale factor 2 from △ABC to △DEF. Which statement must be true?
Answer: D
Corresponding sides are multiplied by the scale factor and corresponding angles are equal. Choice B doubles the angle, which a dilation never does. Choice C pairs ∠E with ∠A, but ∠E corresponds to ∠B. Choice A adds the scale factor.
Question 15 of 20 · Short Answer
△JKL has J(0, 0), K(4, 0), L(0, 6), and △MNV has M(1, 1), N(3, 1), V(1, 4). Decide whether the triangles are similar. If they are, give a sequence of transformations and verify it.
Similar. Dilate by 1/2 about the origin: J(0, 0), K(2, 0), L(0, 3). Translate by (1, 1): (1, 1), (3, 1), (1, 4), which are M, N and V. The scale factor is 1/2: MN = 2 = (1/2) · 4 and MV = 3 = (1/2) · 6.
Question 16 of 20 · Short Answer
Parallelogram ABCD has A(0, 0), B(2, 0), C(3, 1), D(1, 1). Parallelogram EFGH has E(0, 0), F(6, 0), G(9, 3), H(3, 3), and parallelogram WXYZ has W(0, 0), X(6, 0), Y(7, 3), Z(1, 3). Which one is similar to ABCD? Justify both answers.
EFGH is similar: the dilation by 3 about the origin maps A, B, C, D to E, F, G, H exactly. WXYZ is not: WX/AB = 6/2 = 3, but the slanted sides have lengths √10 and √2, a ratio of √5 ≈ 2.24. The angles also differ (the slanted side has slope 1 in ABCD and slope 3 in WXYZ), so no similarity transformation can map ABCD onto WXYZ.
Question 17 of 20 · Short Answer
Explain, using similarity transformations, why △ABC ~ △DEF (with A ↔ D, B ↔ E, C ↔ F) means that ∠B ≅ ∠E and AB/DE = BC/EF.
Sample answer: △ABC ~ △DEF means a sequence of rigid motions and dilations maps A to D, B to E and C to F. Each transformation keeps angle measures, so the image of ∠B, which is ∠E, has the same measure. Rigid motions keep lengths and each dilation multiplies every length by its scale factor, so every length is multiplied by the same total factor k: DE = k · AB and EF = k · BC. Then DE/AB = EF/BC = k, which is the same as AB/DE = BC/EF.
Question 18 of 20 · Short Answer
△ABC ~ △RST with A ↔ R, B ↔ S and C ↔ T. AB = 12, BC = 15, AC = 9 and RS = 8. Find the scale factor from △ABC to △RST, ST and RT.
k = RS/AB = 8/12 = 2/3. ST = (2/3) · 15 = 10 and RT = (2/3) · 9 = 6.
Question 19 of 20 · Short Answer
Use similarity transformations to explain why any two equilateral triangles are similar.
Sample answer: let the triangles have sides s and t. Dilate the first triangle by t/s. Every side becomes t, and every angle is still 60°. Now the image and the second triangle have the same three side lengths, so they are congruent, and a rigid motion maps one onto the other. The dilation followed by that rigid motion maps the first triangle onto the second, so they are similar.
Question 20 of 20 · Short Answer
△ABC has A(1, 2), B(3, 2), C(1, 5), and △DEF has D(-2, -4), E(-6, -4), F(-2, -10). Describe a sequence of transformations that maps △ABC onto △DEF, state the scale factor, and name the angle of △DEF that corresponds to ∠A.
Sample answer: dilate by 2 about the origin, giving (2, 4), (6, 4), (2, 10), then rotate 180° about the origin, giving (-2, -4), (-6, -4), (-2, -10). The scale factor is 2, and ∠D corresponds to ∠A; both are right angles, since AB is horizontal and AC is vertical, and the same is true of DE and DF.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does HSG.SRT.A.2 mean?
HSG.SRT.A.2 means students decide whether two figures are similar by looking for a sequence of rigid motions and dilations between them, and explain why, for triangles, that is the same as having equal corresponding angles and proportional corresponding sides. The definition comes first; the angle and side facts are consequences of it.
Is HSG.SRT.A.2 taught in Geometry?
Yes: it is taught in high school Geometry, in the similarity unit, right after dilations (HSG.SRT.A.1). It extends grade 8 work (8.G.A.4), where students described sequences that exhibit the similarity between two figures, by asking for explanations about triangles.
What is a similarity transformation?
A similarity transformation is a rigid motion (translation, rotation or reflection), a dilation, or any sequence of them. Rigid motions keep size and shape, and dilations keep shape while scaling size. A congruence is a special similarity transformation with total scale factor 1.
How do you prove two figures are similar with transformations?
You name a specific sequence of transformations, in order, and show that it maps every vertex of the first figure onto the matching vertex of the second. With coordinates this is a calculation; with paper figures students use tracing paper and a ruler. A common shortcut is to dilate first by the ratio of corresponding sides and then find the rigid motion.
How do you show that two figures are not similar?
You show that no sequence can work. The usual reason is that two pairs of corresponding sides have different ratios, since every similarity transformation multiplies all lengths by one factor. Another reason is a pair of corresponding angles with different measures, since no similarity transformation changes an angle.
Why do similar triangles have equal angles and proportional sides?
Because each transformation in the sequence keeps angle measures, and each multiplies all lengths by the same factor (1 for rigid motions, k for a dilation). The image of each angle is the corresponding angle, and the image of each side is the corresponding side multiplied by the product of the scale factors.
Are congruent figures also similar?
Yes. Congruent figures are related by a rigid motion, which is a similarity transformation with scale factor 1. So every pair of congruent figures is similar, but similar figures are congruent only when the scale factor is 1.
Are equal angles enough to show that two figures are similar?
Not in general: a square and a 2 by 5 rectangle both have four right angles but are not similar. Triangles are special, and HSG.SRT.A.3 uses this standard to prove that two pairs of equal angles are enough for triangles (the AA criterion). In HSG.SRT.A.2 itself, students should check both angles and side ratios.
Does the order of the transformations matter?
Often it does. Dilating by 3 about the origin and then translating by (2, -1) is not the same as translating first and then dilating, because the dilation also triples the translation. Students should state the order they used and verify the final image. Different sequences can still map one figure onto the same image.
Where does HSG.SRT.A.2 show up later?
Similar triangles are used in HSG.SRT.B.5 to solve problems and prove relationships, in right triangle trigonometry (HSG.SRT.C.6), and in HSG.C.A.1, where students prove that all circles are similar. Problems about similar triangles also appear in the Geometry and Trigonometry domain of the digital SAT.
07
Related Standards
6 standards
These standards connect to HSG.SRT.A.2: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
8.G.A.4Prerequisite
Understand similar figures as images under rigid motions and dilations
