Chapter 5 - Congruence & Triangles - Methods for Showing Triangles to be Congruent

Sep 24, 2026

Uploaded by Levi

Chapter 5 focuses on identifying and proving triangle congruence using multiple standard criteria. It explains how to match congruent figures through corresponding parts and introduces CPCTC. The lesson compares common congruence methods, including SSS, SAS, ASA, and AAS, and highlights the roles of included and non-included sides or angles. Worked examples guide determining corresponding angles and sides, writing congruence statements, and constructing two-column proofs with shared and vertical angles.

LanguageEnglish

CategoryExam

FormatPDF

Pages21

Reading Time53 min

AccessFree to download

5.1 Congruence & Triangles

Methods for Showing Triangles to be Congruent

Congruent Parts of Congruent Triangles

Determine Whether Triangles are Congruent

5.2 Proving Triangles Are Congruent: SSS & SAS

Side-Side-Side Congruence Postulate (SSS)

The “Included” Angle

Side-Angle-Side Congruence Postulate (SAS)

Writing Proofs

5.3 Proving Triangles Are Congruent: ASA & AAS

The “Included” Side

Angle-Side-Angle Congruence Postulate

The “Non-Included” Side

Chapter 5: Congruence & Triangles Methods for Showing Triangles to be Congruent SSS side-side-side right SAS side-angle-side AAS angle-angle-side ASA angle-side-angle HL angle-hypotenuse-leg 5.1 CONGRUENCE & TRIANGLES Goal: Identify congruent triangles and corresponding parts ∂ Congruence 􀂾 Corresponding parts— Key Vocabulary: 􀁸 Corresponding parts 􀁸 Congruent figures 􀂾 Figures are congruent if 􀂃 CPCTC— ∂ Congruent Parts of Congruent Triangles 􀂃 In the diagram below, triangle XYZ is congruent to triangle MNP. 􀂃 Written as: 􀁕XYZ 􀀣 􀁕MNP. Congruencstatemente 􀁸 The notation shows the corresponding vertices and thus the corresponding angles & sides. Corresponding Angles Corresponding Sides Examples: Determine whether the angles or sides are corresponding angles, corresponding sides, or neither. Given that 􀁕XYZ 􀀣 􀁕EFD, determine the congruent side or angle that corresponds to the side or angle. Example 9: Given that 􀁕STU 􀀣 􀁕YXZ, list all corresponding congruent parts. Angles Example 10: Given 􀁕ABC 􀀣 􀁕DEF, find the length of and m􀂑B. Example 11: Which congruence statement is correct? Why? Sides ∂ Determine Whether Triangles are Congruent 􀂾 In the figure, || and || 􀂃 Determine whether two triangles are congruent. If they are, write a congruence statement. 􀁸 Remember that when parallel lines are cut by a transversal, the alternate interior angles are congruent. 􀂃 Start by listing the side(s) &/or angles you know to be congruent. 􀂃 Then list any info you can deduce from the figure. Be on the lookout for vertical angles and shared sides of both triangles. Examples: Determining Congruence Determine whether the triangles are congruent. If they are, write a congruence statement. 12. 13. Example 14: Given that 􀁕FGH􀀣 􀁕TSR, determine which congruence statement does not describe the triangles. a. 􀁕HGF 􀀣 􀁕RST b. 􀁕FHG 􀀣 􀁕TRS c. 􀁕GFH􀀣 􀁕SRT Examples: Geo-Gebra 15. If 􀁕RST 􀀣 􀁕XYZ, find the value of n. To do this problem you must identify the angle congruent to 􀂑Y. 16. If 􀁕RQP 􀀣 􀁕ONM, find the value of m. To do this problem you must identify the side congruent to 17. Refer to Jillian’s work below and answer the questions that follow. a. How does Jillian know that 􀂑BAC 􀀣 􀂑DAE? b. Is 􀁕ABC 􀀣 􀁕ADE? Explain your reasoning. 5.2 PROVING TRIANGLES ARE CONGRUENT: SSS & SAS Goal: Show triangles are congruent using SSS & SAS. Accessing Prior Knowledge: Pre-Proof Warm-Ups [http://www.regentsprep.org/Regents/math/geometry/GP3/preproof.htm](http://www.regentsprep.org/Regents/math/geometry/GP3/preproof.htm) ∂ Side – Side – Side Congruence Postulate (SSS) 􀂾 If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent. 􀂃 Sides: Examples: Using the SSS Congruence Postulate Does the diagram give enough information to use the SSS Congruence Postulate? Explain your reasoning. 1. 2. List those angles &/or sides you know to be congruent: List those angles &/or sides you know to be congruent: ∂ The “Included” Angle 􀂾 In 􀁕ABC, is the included angle between sides . Example 3: Identifying the Included Angle Use the diagram to name the included angle between the two given sides. a. b. c. d. e. ∂ Side – Angle – Side Congruence Postulate (SAS) 􀂾 If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent. 􀂃 Side: Angle: Side: Examples: Using the SAS Congruence Postulate Does the diagram give enough information to use the SAS Congruence Postulate? Explain your reasoning. 4. 5. List those angles &/or sides you know to be congruent: List those angles &/or sides you know to be congruent: 6. What additional information is needed to prove the two triangles are congruent by the SAS Congruence Postulate? ∂ Writing Proofs 􀂾 Proof– 􀂃 The proof begins with the

FAQs

How do you show two triangles are congruent using corresponding parts?

Identify the matching corresponding vertices, then use corresponding angles and corresponding sides to form a congruence statement. If triangles are congruent, CPCTC confirms corresponding parts are congruent.

When using SSS, what information is needed?

You need three sides of one triangle congruent to three sides of a second triangle. If all three side pairs match, the triangles are congruent by the SSS Congruence Postulate.

What is the difference between the included and non-included side in ASA and AAS?

For ASA, the side between the two angles is the included side. For AAS, the side between the two angles is not included; instead, a non-included side is used along with two congruent angles and their corresponding side condition.

Sep 24, 2026

Uploaded by Levi