HSG.SRT.B.5Common CoreMathGeometryGrades 9-12

HSG.SRT.B.5: Using Congruence and Similarity Criteria in Problems and Proofs

In plain English: HSG.SRT.B.5 is the Common Core geometry standard that asks students to use the triangle congruence criteria (SSS, SAS, ASA, AAS, HL) and similarity criteria (AA, SAS, SSS) to find unknown lengths and angles and to prove relationships in figures. The key move is to find a pair of triangles hidden in the figure. It is usually taught in high school Geometry.

Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Common Core State Standards for Mathematics · Domain: Similarity, Right Triangles, and Trigonometry (SRT) · Cluster: Prove theorems involving similarity
Also written as HSG-SRT.B.5 or G-SRT.5 · Official standard

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Lesson Plan

70 min

Overview

By this point students know the congruence criteria (SSS, SAS, ASA, AAS and HL for right triangles) and the similarity criteria (AA, SAS similarity and SSS similarity). This lesson is about using them. In a real figure the triangles overlap, share a side or sit inside a trapezoid, and the given information is spread across parallel lines, vertical angles and right angles. Students learn to find the pair of triangles that matters, to name the criterion, and then to use the conclusion: congruent parts are equal, and corresponding sides of similar triangles are proportional.

The lesson has two strands that the standard names. In the problem strand, students find heights with shadows and mirrors, find lengths in nested and overlapping triangles, and use the altitude to the hypotenuse. In the proof strand, students write short proofs of relationships such as "the diagonals of a trapezoid cut each other in the same ratio" and "the diagonal of a kite bisects its vertex angles."

Learning Objectives

By the end of this lesson, students will be able to:

Choose the congruence criterion (SSS, SAS, ASA, AAS or HL) or similarity criterion (AA, SAS or SSS similarity) that the given information supports, and reject SSA and AAA as congruence tests
Find unknown lengths and angles by proving two triangles congruent or similar and then using corresponding parts
Solve indirect-measurement problems with shadows, mirrors and the altitude to the hypotenuse
Write a two-column or paragraph proof of a relationship in a figure by first proving a pair of triangles congruent or similar

Prior Knowledge Required

Students should already be comfortable with:

The triangle congruence criteria ASA, SAS and SSS and why they work HSG.CO.B.8
The AA criterion for similar triangles HSG.SRT.A.3
Angle relationships with parallel lines and vertical angles HSG.CO.C.9
Solving proportions 7.RP.A.2

Lesson Procedure

70-70 minutes of class time across 5 phases.

Warm-Up10 minutes
Sketch four pairs of triangles on the board, each with tick marks and arcs: (1) two sides and the included angle marked; (2) two angles and a non-included side marked; (3) all three angles marked and no sides; (4) two sides and an angle that is not between them.
Warm-Up Prompt
"For each pair, decide: congruent, similar but maybe not congruent, or not enough information. Name the criterion you used. Which pair is a trap?"
Expected answers: (1) congruent by SAS, (2) congruent by AAS, (3) similar by AA but not necessarily congruent, (4) not enough information, because SSA is not a criterion. Have a student build pair 4 with patty paper to show two different triangles with the same SSA parts. This sets up the day's routine: before using any conclusion, name the criterion that earns it.

Direct Instruction25 minutes

Triangle criteria and what each one lets you conclude
Criterion	What you need	What you may conclude
SSS, SAS, ASA, AAS	Three matching parts, including at least one pair of sides	The triangles are congruent, so every pair of corresponding parts is congruent (CPCTC)
HL	Two right triangles with congruent hypotenuses and one pair of congruent legs	The right triangles are congruent
AA	Two pairs of congruent angles	The triangles are similar, so corresponding sides are proportional
SAS similarity	Two pairs of sides in the same ratio and the included angles congruent	The triangles are similar
SSS similarity	All three pairs of sides in the same ratio	The triangles are similar

Model a four-step routine for every problem and proof, and keep it posted:

Find the triangles. Highlight the two triangles that contain the unknown or the parts in the statement to prove. Redraw overlapping triangles side by side.
Collect matching parts. Mark given parts, then add parts that come free: a shared side (reflexive property), vertical angles, alternate interior angles from parallel lines, right angles.
Name the criterion. Write the congruence or similarity statement with the vertices in matching order.
Use the conclusion. For congruent triangles, corresponding parts are congruent. For similar triangles, write one proportion from corresponding sides and solve.

Work these five examples with the class. Examples 3 and 5 match Diagrams 1 and 2.

Similarity problem: shadows (AA)
A student 1.6 m tall casts a 2.4 m shadow. At the same moment a flagpole casts a 13.5 m shadow. How tall is the flagpole?
Equation: The sun's rays make the same angle with the ground, and both stand at right angles to the ground, so the triangles are similar by AA: 1.6/2.4 = h/13.5, h = 9 m
Similarity problem: nested triangles (AA)
In △ABC, D is on AB and E is on AC with DE ∥ BC. AD = 6, DB = 4 and DE = 9. Find BC.
Equation: ∠ADE ≅ ∠ABC (corresponding angles) and ∠A is shared, so △ADE ~ △ABC by AA. Use the whole side: 6/10 = 9/BC, so BC = 15
Similarity problem: altitude to the hypotenuse (Diagram 1)
In right △ACB with the right angle at C, altitude CD splits hypotenuse AB into AD = 4 and DB = 9. Find CD and both legs.
Equation: △ADC ~ △CDB by AA, so 4/CD = CD/9 and CD = 6. Also AC² = AD · AB = 52 and BC² = DB · AB = 117, so AC = 2√13 ≈ 7.21 and BC = 3√13 ≈ 10.82
Congruence proof and problem: kite (SSS)
Kite ABCD has AB = AD and CB = CD, with ∠BAD = 70°. Prove that diagonal AC bisects ∠BAD, then find ∠BAC.
Equation: AB = AD, CB = CD and AC = AC (shared), so △ABC ≅ △ADC by SSS. By CPCTC, ∠BAC ≅ ∠DAC, so AC bisects ∠BAD and ∠BAC = 70°/2 = 35°
Similarity proof and problem: trapezoid diagonals (Diagram 2)
Trapezoid ABCD has AB ∥ DC, AB = 12 and DC = 8. Its diagonals meet at E, and AE = 9. Prove that AE/EC = AB/DC, then find EC.
Equation: ∠BAE ≅ ∠DCE and ∠ABE ≅ ∠CDE (alternate interior angles), so △ABE ~ △CDE by AA and AE/CE = AB/CD. Then 9/EC = 12/8, so EC = 6

Guided Practice15 minutes
Pairs complete one proof and one problem, then trade papers and check each other's criterion.
- Proof. In △PQR, PQ = PR. Let M be the midpoint of QR and draw PM. Prove that ∠Q ≅ ∠R. (PQ = PR, QM = RM and PM = PM, so △PQM ≅ △PRM by SSS, and ∠Q ≅ ∠R by CPCTC.)
- Problem. A student whose eyes are 1.5 m above the ground places a mirror flat on the ground 14 m from the base of a building. She steps back until she sees the top of the building in the mirror, which happens when she is 2 m from the mirror. How tall is the building? (The angles of incidence and reflection are equal and both people and building are perpendicular to the ground, so the triangles are similar by AA: 1.5/2 = h/14, h = 10.5 m.)
Listen for pairs who write the proportion with the wrong sides matched. Ask them to point to the corresponding angle opposite each side before they set up the ratio.
Independent Practice15 minutes
Students work alone:
- In parallelogram WXYZ, diagonals WY and XZ meet at K. Prove that K is the midpoint of both diagonals. (∠KWX ≅ ∠KYZ and ∠KXW ≅ ∠KZY are alternate interior angles, and WX = YZ, so △WKX ≅ △YKZ by ASA; then WK = YK and XK = ZK by CPCTC.)
- In △PQR, S is on PQ and T is on PR with ST ∥ QR. PS = 5, SQ = 10 and ST = 7. Find QR. (△PST ~ △PQR by AA, and 5/15 = 7/QR, so QR = 21.)
- One triangle has sides 8 cm and 12 cm with a 40° angle between them, and its third side is 7.8 cm. A second triangle has sides 12 cm and 18 cm with a 40° angle between them. Are the triangles similar? Find the second triangle's third side. (8/12 = 12/18 = 2/3 and the included angles match, so they are similar by SAS similarity; the third side is 7.8 · 1.5 = 11.7 cm.)
Closure5 minutes
Exit ticket: (1) Draw two triangles that share a side and have one more pair of congruent sides. What third fact would make them congruent, and by which criterion? (2) In one sentence, explain the difference between what you may conclude from congruent triangles and from similar triangles.

Differentiation Strategies

For Struggling Students

Give a two-color highlighting routine: one color for the first triangle, a second for the other, so shared sides and angles show up as overlaps
Provide a proof frame with the four routine steps as headings and blank lines for statements and reasons
Have students redraw overlapping triangles separately and turn them to the same orientation before writing a proportion

For Advanced Students

Prove that in any right triangle, the altitude to the hypotenuse is the geometric mean of the two segments it creates, using letters instead of numbers
Ask students to write a single proof that uses both a congruence and a similarity: for example, show that the midpoints of a trapezoid's diagonals lie on a segment parallel to the bases
Challenge: explain why HL works even though it looks like SSA, using the Pythagorean theorem to find the third side

Assessment Guidance

What to Look For

Check that every conclusion about sides or angles comes after a named criterion, and that the congruence or similarity statement lists vertices in matching order. In problems, look for a proportion that pairs sides opposite the same angles and for units in the final answer. In proofs, look for the parts that come free in the figure (a shared side, vertical angles, alternate interior angles) with the reason written out. A student who uses SSA or AAA as a congruence test needs a patty-paper counterexample.

Classroom Activities

3 Activities

Criterion Sort

15 minGroups of 3

Each group receives 10 cards. Every card describes a pair of triangles with the given information. Groups sort the cards into three piles: congruent (write the criterion), similar but not necessarily congruent (write the criterion), and not enough information (write a reason). Then each group picks one card from the last pile and adds one fact that moves it to another pile.

Card Set

Card 1: AB = DE, BC = EF, CA = FD (congruent, SSS)
Card 2: ∠A ≅ ∠D, ∠B ≅ ∠E, AB = DE (congruent, ASA)
Card 3: ∠A ≅ ∠D, ∠B ≅ ∠E, BC = EF (congruent, AAS)
Card 4: ∠C and ∠F are right angles, AB = DE, AC = DF (congruent, HL)
Card 5: ∠A = 52°, ∠B = 71°, ∠D = 71°, ∠E = 57° (similar, AA, since ∠C = 57° and ∠F = 52°)
Card 6: sides 4, 6, 9 and sides 10, 15, 22.5 (similar, SSS similarity, ratio 2.5)
Card 7: AB = 5, AC = 8, ∠A = 64° and DE = 7.5, DF = 12, ∠D = 64° (similar, SAS similarity)
Card 8: AB = DE, AC = DF, ∠B ≅ ∠E, where ∠B is not between the marked sides (not enough: SSA)
Card 9: all three pairs of angles congruent, no sides given (similar only)
Card 10: sides 3, 5, 6 and sides 6, 10, 11 (not similar: 11/6 is not 2)

Modification for Distance Learning

Put the cards on a shared slide with three labeled columns. Each group drags the cards into place and types the criterion or reason in a text box next to each card.

Mirror Method Measurement

20 minGroups of 3

Groups measure the height of something they cannot reach, such as a gym wall, a basketball backboard or a light pole, with a small mirror and a tape measure. They then write the similarity argument that makes the measurement valid.

Procedure

Tape a small mirror flat on the floor a measured distance from the base of the object
One student backs away from the mirror until the top of the object appears at the center of the mirror; a second student measures that student's distance from the mirror and eye height
Record the three measurements and solve the proportion (eye height)/(student-to-mirror distance) = (object height)/(mirror-to-object distance)
Sample data from a trial run: eye height 1.62 m, student 1.8 m from the mirror, mirror 6.3 m from the wall, so the wall is 1.62 · 6.3/1.8 = 5.67 m tall

Write the Justification

Name the two triangles and write the similarity statement in matching order
Give a reason for each pair of congruent angles (right angles with the floor; equal angles of incidence and reflection)
List two sources of measurement error and say whether each would make the answer too large or too small

Variation: Shadow Method

On a sunny day, measure a student's height and shadow and the shadow of a tree or flagpole at the same time, then compare the two methods for the same object.

Proof Relay

20 minGroups of 3

Each proof is split into three legs. Student 1 finds and highlights the triangles, Student 2 collects the matching parts with reasons and names the criterion, and Student 3 writes the conclusion. The paper is passed after each leg, and the group rotates roles for the next proof.

Proof Cards

Proof A: In isosceles △JKL with JK = JL, the altitude JN to KL bisects KL. (Right △JNK and △JNL share leg JN and have congruent hypotenuses, so they are congruent by HL; KN = LN by CPCTC.)
Proof B: In △FGH, M is the midpoint of FG and N is the midpoint of FH. Then MN ∥ GH and MN = GH/2. (FM/FG = FN/FH = 1/2 and ∠F is shared, so △FMN ~ △FGH by SAS similarity; then ∠FMN ≅ ∠FGH gives MN ∥ GH as corresponding angles, and MN/GH = 1/2.)
Proof C: The diagonals of rhombus FGHI are perpendicular. (The diagonals bisect each other at O, so FO = HO; FG = HG and GO = GO give △FOG ≅ △HOG by SSS; the two angles at O are congruent and form a linear pair, so each is 90°.)

Debrief

Which proofs used a criterion for congruence, and which used similarity?
In each proof, which matching part came from the figure rather than from the given information?

Diagrams & Visual Aids

2 diagrams

Diagram 1: Three Similar Triangles from One Altitude

In right △ACB, the altitude CD to the hypotenuse makes two smaller right triangles. Each shares an acute angle with △ACB, so all three triangles are similar by AA, and the proportion AD/CD = CD/DB gives CD² = AD · DB. Drawn to scale for Example 3.

Diagram 2: Similar Triangles Inside a Trapezoid

The parallel bases make two pairs of congruent alternate interior angles, so △ABE ~ △CDE by AA and the diagonals cut each other in the ratio AB : DC. Drawn to scale for Example 5.

Homework Assignment

~30 min

HSG.SRT.B.5 Homework: Congruence and Similarity in Problems and Proofs

Directions: For every problem, draw and label a figure, name the two triangles you use, and state the criterion (SSS, SAS, ASA, AAS, HL, AA, SAS similarity or SSS similarity) before using any corresponding parts. Give exact answers where possible and decimals to the nearest hundredth.

Part 1: Solving Problems (Problems 1-3)

A hiker 1.8 m tall casts a 1.2 m shadow. At the same time, a pine tree casts a 7.4 m shadow. Explain why the two triangles are similar, and find the height of the tree.
In right △XYZ with the right angle at Z, altitude ZW divides hypotenuse XY into XW = 5 and WY = 20. Find ZW, XZ and YZ. Name the similar triangles you used.
Segments AB and CD bisect each other at M. AC = 14 cm and ∠CAM = 38°. Prove that △AMC ≅ △BMD, then find BD and ∠DBM.

Part 2: Proving Relationships (Problems 4-6)

△ABC ~ △DEF with scale factor k = DE/AB. Altitude AG is drawn to BC and altitude DH is drawn to EF. Prove that DH/AG = k.
In △ABC, point D lies on AC so that ∠ABD ≅ ∠C. Prove that △ABD ~ △ACB and that AB² = AD · AC. Then find AB when AD = 4 and AC = 9.
In △ABC, segment AD bisects ∠BAC and is perpendicular to BC at D. Prove that △ABC is isosceles with AB = AC.

Rubric

Criterion	Full Credit (2 pts)	Partial Credit (1 pt)	No Credit (0 pts)
Choice of Triangles and Criterion	Correct pair of triangles and a valid criterion named for every problem	Correct triangles but one criterion missing or misnamed	No criterion, or SSA or AAA used as a congruence test
Reasons in Proofs	Every statement has a reason, including parts that come from the figure	One or two missing reasons	Statements without reasons
Correspondence	Vertices listed in matching order and proportions pair corresponding sides	One mismatched pair	Proportions pair non-corresponding sides
Accuracy	All lengths and angles correct with units	One calculation error	Several errors

Quiz: 20 Questions

Interactive, with answers

Instructions

Choose an answer for each multiple-choice question, then read the explanation. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again. For short-answer questions, work on paper and then open the answer.

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
△ABC and △DEF have AB = DE = 9 cm, BC = EF = 6 cm and ∠A ≅ ∠D = 35°. Which statement is correct?
Question 2 of 20 · Multiple Choice
△GHI has angles of 48° and 67°. △JKL has angles of 67° and 65°. What can you conclude?
Question 3 of 20 · Multiple Choice
One triangle has sides 6, 9 and 12. Another has sides 8, 12 and 16. Which statement is true?
Question 4 of 20 · Multiple Choice
△RST has RS = 5, RT = 7 and ∠R = 50°. △UVW has UV = 15, UW = 21 and ∠U = 50°. If ST = 6.2, find VW.
Question 5 of 20 · Multiple Choice
A 3 m pole casts a 4.5 m shadow. At the same time, a building casts a 27 m shadow. How tall is the building?
Question 6 of 20 · Multiple Choice
In △ABC, D is on AB and E is on AC with DE ∥ BC. AD = 4, AB = 10 and DE = 5. Find BC.
Question 7 of 20 · Multiple Choice
In a right triangle, the altitude to the hypotenuse divides the hypotenuse into segments of 2 and 8. How long is the altitude?
Question 8 of 20 · Multiple Choice
You know ∠A ≅ ∠D and AB ≅ DE in △ABC and △DEF. Which extra fact proves the triangles congruent by ASA?
Question 9 of 20 · Multiple Choice
A proof has shown △KLM ≅ △NOP. Which statement may be written next, with the reason CPCTC?
Question 10 of 20 · Multiple Choice
To prove that opposite sides of parallelogram ABCD are congruent, a student draws diagonal AC and proves △ABC ≅ △CDA by ASA. Which angle pairs does the student use, and why are they congruent?
Question 11 of 20 · Multiple Choice
Two right triangles have congruent hypotenuses of 13 cm, and each has a leg of 5 cm. Which criterion proves them congruent?
Question 12 of 20 · Multiple Choice
A surveyor's eyes are 1.6 m above the ground. She stands 2.5 m from a mirror lying flat on the ground and sees the top of a tower in it. The mirror is 20 m from the base of the tower. How tall is the tower?
Question 13 of 20 · Multiple Choice
Two triangular roof trusses both have angles of 30°, 60° and 90°. The longest side of one is 4 m, and the longest side of the other is 6 m. A student says the trusses are congruent because all three pairs of angles match. What is the best response?
Question 14 of 20 · Multiple Choice
In △PQR, point S lies on PR so that ∠PQS ≅ ∠R. Which similarity statement is correct?
Question 15 of 20 · Short Answer
Rectangle ABCD has diagonals AC and BD. Prove that AC = BD.
Question 16 of 20 · Short Answer
Segments AC and BD cross at E, with AE = 6, EC = 4, BE = 9, ED = 6 and CD = 5. Prove that △ABE ~ △CDE, then prove that AB ∥ CD and find AB.
Question 17 of 20 · Short Answer
To find the width AB of a river, a surveyor stands at point B on her side of the river, directly across from a tree at A, and marks point E farther along the bank. The line from the tree through E meets a stake at D. She places stake C so that ∠C and ∠B are right angles and B, E and C are collinear. She measures BE = 30 m, CE = 12 m and CD = 16 m. Name the criterion that makes the triangles similar and find AB.
Question 18 of 20 · Short Answer
In isosceles △XYZ, XY = XZ. M is the midpoint of XY and N is the midpoint of XZ. Prove that the medians ZM and YN are congruent.
Question 19 of 20 · Short Answer
△ABC ≅ △DEF, with AB = 3x + 2 and DE = 5x - 8. Find x and AB, and name the reason that lets you set the two expressions equal.
Question 20 of 20 · Short Answer
In △ABC, AB = 12, AC = 9 and BC = 15. Point D is on AB with AD = 3, and point E is on AC with AE = 4. Is DE parallel to BC? Prove which pair of triangles is similar, and find DE.

0 of 20 answered · 0 correct

Frequently Asked Questions

10 Questions

What does HSG.SRT.B.5 mean?

It means students use the triangle congruence and similarity criteria as tools. Given a figure, they find a pair of congruent or similar triangles, name the criterion, and use the result to find a missing length or angle or to prove that some relationship always holds.

Is HSG.SRT.B.5 taught in Geometry or Algebra 2?

It is usually taught in high school Geometry. It follows the units on rigid motions, congruence and similarity transformations and comes before right-triangle trigonometry, which relies on similar right triangles.

What are the triangle congruence criteria?

SSS, SAS, ASA and AAS work for any triangle, and HL works for right triangles. Each one names three matching parts, including at least one pair of sides, that force the triangles to have the same size and shape.

What are the similarity criteria for triangles?

AA (two pairs of congruent angles), SAS similarity (two pairs of proportional sides with congruent included angles) and SSS similarity (all three pairs of sides in the same ratio). Similar triangles have the same shape, and their corresponding sides are proportional.

Why don't SSA and AAA prove triangles congruent?

Neither one fixes the triangle. With SSA, the side opposite the given angle can swing into two positions, so two different triangles can share the same three parts. With AAA, the shape is fixed but the size is not, so the triangles are similar only.

What is CPCTC?

It stands for "corresponding parts of congruent triangles are congruent." It is the reason you give after a congruence statement when you conclude that a pair of sides or angles is congruent. It may be used only after the triangles have been proved congruent.

How do you find similar triangles in a complicated figure?

Look for parallel lines, which create congruent corresponding or alternate interior angles; for vertical angles where segments cross; for right angles; and for an angle two triangles share. Two of those clues usually give AA. Redrawing the triangles separately in the same orientation makes the correspondence clear.

What are common mistakes on HSG.SRT.B.5 problems?

Writing a proportion with non-corresponding sides, using a piece of a side instead of the whole side in nested triangles, and listing the vertices of a congruence statement in the wrong order. In proofs, a frequent gap is using CPCTC before the triangles are proved congruent.

Where is similarity used in real life?

Surveyors, builders and photographers use it for indirect measurement: heights from shadows or mirrors, distances across rivers, and scale drawings and maps. Right-triangle trigonometry is also built on it, since every right triangle with a given acute angle is similar to every other.

Is this standard on the SAT?

Yes, in part. The digital SAT Geometry and Trigonometry domain includes problems that use congruent and similar triangles to find lengths and angles. Formal written proofs are not tested there, but they are a regular part of Geometry course assessments aligned to this standard.