Chapter 5: Congruent Triangles
Learning Objectives
- Identify corresponding parts of congruent triangles using proper notation
- Apply the SSS, SAS, ASA, AAS, and HL congruence postulates and theorems
- Recognize which shortcut applies (and which do not: AAA, SSA)
- Use CPCTC to prove additional corresponding parts congruent
- Write complete two-column proofs for triangle congruence
- Apply the Isosceles Triangle Theorem and its converse
- Chapter Summary
- Key Terms
5.1 Congruent Figures and Corresponding Parts
Two figures are congruent if they have the same shape and size — one can be mapped onto the other by rigid motions (translations, rotations, reflections) only.
Definition: Congruent Triangles
$\triangle ABC \cong \triangle DEF$ means that all six pairs of corresponding parts are congruent:
- Sides: $AB \cong DE$, $BC \cong EF$, $AC \cong DF$
- Angles: $\angle A \cong \angle D$, $\angle B \cong \angle E$, $\angle C \cong \angle F$
Order matters! The order of vertices in the congruence statement specifies which parts correspond: first vertex ↔ first vertex, second ↔ second, etc.
Example 5.1 — Reading a Congruence Statement
If $\triangle PQR \cong \triangle XYZ$, list all six pairs of congruent parts.
Sides: $PQ \cong XY$, $QR \cong YZ$, $PR \cong XZ$
Angles: $\angle P \cong \angle X$, $\angle Q \cong \angle Y$, $\angle R \cong \angle Z$
5.2 Triangle Congruence Postulates and Theorems
We don't need to verify all six pairs to prove triangles congruent. Five valid shortcuts (and two invalid ones) are shown below:
The Five Congruence Shortcuts
- SSS: If $AB = DE$, $BC = EF$, and $AC = DF$, then $\triangle ABC \cong \triangle DEF$.
- SAS: If $AB = DE$, $\angle B = \angle E$, and $BC = EF$, then $\triangle ABC \cong \triangle DEF$. (Angle must be between the two sides.)
- ASA: If $\angle A = \angle D$, $AB = DE$, and $\angle B = \angle E$, then $\triangle ABC \cong \triangle DEF$. (Side must be between the two angles.)
- AAS: If $\angle A = \angle D$, $\angle B = \angle E$, and $BC = EF$, then $\triangle ABC \cong \triangle DEF$. (Side is not between the angles.)
- HL (Right Triangles only): If the hypotenuse and one leg of a right triangle are congruent to those of another, then the triangles are congruent.
Memory Aid: "SAS means the angle is sandwiched between the two sides." For SSA — remember "a donkey is stubborn" — SSA doesn't work! The angle must be included between the sides for SAS, and the side must be included between the angles for ASA.
Example 5.2 — Identifying the Correct Postulate
Which postulate or theorem proves $\triangle JKL \cong \triangle MNP$?
- $JK = MN$, $KL = NP$, $\angle K = \angle N$: The angle $\angle K$ is between sides $JK$ and $KL$, and $\angle N$ is between $MN$ and $NP$. This is SAS.
- $\angle J = \angle M$, $\angle L = \angle P$, $JL = MP$: Two angles and the side between them. This is ASA.
- $\angle J = \angle M$, $JK = MN$, $KL = NP$: Two angles... wait, we only have one angle. With $JK$ and $KL$ being sides, we actually have SSA — not valid. Need to verify the angle is included.
Right $\triangle ABC$ and $\triangle DEF$ have right angles at $B$ and $E$. If $AC = DF$ (hypotenuses) and $AB = DE$ (legs), which postulate proves congruence?
Show Answer
SSS, SAS, ASA, AAS illustrated — pairs of congruent triangles with color-coded matching parts.
Figure 5.1 — Congruent Triangle Pairs by Different Postulates
5.3 CPCTC: Corresponding Parts of Congruent Triangles Are Congruent
CPCTC
CPCTC stands for "Corresponding Parts of Congruent Triangles Are Congruent." Once you prove two triangles congruent, you can conclude that any pair of corresponding parts are also congruent. CPCTC is used as a reason in the later steps of a proof, after establishing the triangle congruence.
The typical two-column proof structure when using CPCTC:
- Establish congruence of specific sides and angles (given or derived)
- State the triangle congruence (SSS / SAS / ASA / AAS / HL)
- Conclude additional corresponding parts are congruent by CPCTC
Example 5.3 — A Complete Two-Column Proof with CPCTC
Given: $\overline{AB} \cong \overline{CB}$; $B$ is the midpoint of $\overline{AC}$ — wait, let's use a cleaner setup:
Given: $\overline{AB} \cong \overline{CD}$; $\overline{AB} \parallel \overline{CD}$
Prove: $\triangle ABE \cong \triangle DCE$ where $E$ is the intersection of $\overline{AD}$ and $\overline{BC}$
| Statements | Reasons |
|---|---|
| $AB \cong CD$ | Given |
| $AB \parallel CD$ | Given |
| $\angle ABE \cong \angle DCE$ | Alternate interior angles, $AB \parallel CD$, transversal $\overline{BC}$ |
| $\angle BAE \cong \angle CDE$ | Alternate interior angles, $AB \parallel CD$, transversal $\overline{AD}$ |
| $\triangle ABE \cong \triangle DCE$ | AAS |
| $BE \cong CE$ and $AE \cong DE$ | CPCTC |
In the proof above, what additional conclusion can you draw from CPCTC? (Hint: what does $AE \cong DE$ and $BE \cong CE$ tell you about point $E$?)
Show Answer
5.4 Isosceles Triangle Theorem
Isosceles Triangle Theorem and Converse
- Theorem: If two sides of a triangle are congruent, then the angles opposite those sides are congruent. (Base angles of an isosceles triangle are congruent.)
- Converse: If two angles of a triangle are congruent, then the sides opposite those angles are congruent.
- Equilateral ↔ Equiangular: A triangle is equilateral if and only if it is equiangular (all angles = 60°).
Example 5.4 — Isosceles Triangle
In $\triangle ABC$, $AB = AC = 7$ and $\angle ABC = 52°$. Find $\angle BAC$.
By the Isosceles Triangle Theorem, $\angle ABC = \angle ACB = 52°$.
$\angle BAC = 180° - 52° - 52° = \mathbf{76°}$
Isosceles triangle with vertex angle slider — observe that the base angles remain equal as the shape changes.
Figure 5.2 — Isosceles Triangle: Base Angles Always Equal
SSS vs. SAS comparison — both postulate pairs produce the same unique triangle shape, confirming congruence.
Figure 5.3 — SSS and SAS Produce Unique (Congruent) Triangles
Practice Problems
$\triangle ABC \cong \triangle PQR$. If $AB = 3x + 2$ and $PQ = 5x - 6$, find $AB$.
Show Solution
For each set of given info, identify the congruence postulate or state "not enough":
(a) Two sides and the included angle
(b) Two angles and any side
(c) Two sides and a non-included angle
(d) Three angles only
Show Solution
$\overline{WX}$ bisects $\angle VWY$ and $\overline{WX} \perp \overline{VY}$. Prove $\triangle VWX \cong \triangle YWX$. What postulate applies?
Show Solution
In isosceles $\triangle KLM$ with $KL = KM$, $\angle L = 3x + 10$ and $\angle M = 5x - 4$. Find all angles.
Show Solution
In the diagram, $\overline{AC}$ is a common side of $\triangle ABC$ and $\triangle ACD$. Given $AB \cong CD$ and $BC \cong AD$, prove the triangles congruent. (Which postulate?)
Show Solution
$\triangle ABC \cong \triangle DEF$ (proved by ASA). Can you conclude $BC \cong EF$? What about $\angle A \cong \angle D$? State your reasoning.
Show Solution
Two-column proof: Given $M$ is the midpoint of $\overline{AB}$ and $M$ is the midpoint of $\overline{CD}$. Prove $\triangle AMC \cong \triangle BMD$.
Show Solution
1. $M$ is midpoint of $AB$ / Given
2. $AM \cong MB$ / Definition of midpoint
3. $M$ is midpoint of $CD$ / Given
4. $CM \cong MD$ / Definition of midpoint
5. $\angle AMC \cong \angle BMD$ / Vertical angles
6. $\triangle AMC \cong \triangle BMD$ / SAS $\square$
In right $\triangle RST$ (right angle at $S$) and right $\triangle UVW$ (right angle at $V$), $RT = UW$ and $RS = UV$. Prove $\triangle RST \cong \triangle UVW$ and find $\angle T$ if $\angle R = 35°$.
Show Solution
📋 Chapter Summary
Congruence Postulates and Theorems
If three sides of one triangle are congruent to three sides of another, the triangles are congruent.
Two sides and the included angle (between those sides) are congruent. The angle must be between the two sides.
Two angles and the included side are congruent. The side must be between the two angles.
Two angles and a non-included side are congruent. Different from ASA — the side is not between the angles.
For right triangles only: hypotenuse and one leg are congruent. The right angle is already known.
"Corresponding Parts of Congruent Triangles are Congruent" — used after proving triangles congruent to conclude other parts are equal.
Isosceles Triangle Theorem
- Base angles of an isosceles triangle are congruent
- Converse: if two angles of a triangle are congruent, the sides opposite them are congruent
- The angle bisector from the vertex angle of an isosceles triangle is also the perpendicular bisector of the base