Congruent Triangles

📐 Congruent Triangles

Part 1 of 5 — What Congruence Means & Correspondence

Topics in This Part

🔑 Key Concept: Two triangles are congruent when they have exactly the same size and shape — one can be slid, turned, or flipped to land perfectly on the other. The whole unit is about proving this without measuring all six parts.

Congruent Figures & Rigid Motions

Two figures are congruent ($\cong$) if a sequence of rigid motions maps one exactly onto the other. The three rigid motions are:

Because rigid motions never stretch or shrink, congruent triangles have:

• all three pairs of sides equal in length, and
• all three pairs of angles equal in measure.

That's six matching pieces in total — three sides and three angles.

💡 "Equal" describes measures (numbers); "congruent" describes figures (segments, angles, triangles). We write $AB = CD$ for lengths but $\overline{AB} \cong \overline{CD}$ for the segments themselves.

Count the Parts 🧮

A triangle has three sides and three angles.

1) How many pairs of sides must match for two triangles to be congruent? 2) How many pairs of angles must match? 3) In total, how many corresponding parts are congruent in a pair of congruent triangles?

Correspondence: Order Matters

When we write a congruence statement, the order of the letters tells you which parts match. If

$\triangle ABC \cong \triangle DEF$

then the vertices correspond in order:

$A \leftrightarrow D, \quad B \leftrightarrow E, \quad C \leftrightarrow F$

From that single statement you can read off all six congruences:

⚠️ Common Mistake: $\triangle ABC \cong \triangle DEF$ and $\triangle ABC \cong \triangle EFD$ are not the same statement. The letters must line up matching part to matching part.

Concept Check 🎯

The Big Payoff: CPCTC

Once you prove two triangles congruent, you instantly know all six parts match. That principle has a famous name:

🔑 CPCTC — Corresponding Parts of Congruent Triangles are Congruent.

This is the reason we care about proving congruence. Often a problem wants just one segment or one angle, and the strategy is:

1. Prove the two triangles containing those parts are congruent.
2. Use CPCTC to conclude the specific part you wanted is congruent.

The next three parts give you the shortcuts (SSS, SAS, ASA, AAS, HL) that prove congruence from only three pieces — so you never have to check all six.

Match the Parts 🔽

Given $\triangle MNO \cong \triangle XYZ$, choose the corresponding part for each.

📐 Congruent Triangles

Part 2 of 5 — The SSS & SAS Postulates

🔑 The Idea: You don't need all six parts. Just three of the right ones lock a triangle's size and shape. The first two shortcuts are SSS (three sides) and SAS (two sides and the angle between them).

Side-Side-Side (SSS)

🔑 SSS Postulate: If the three sides of one triangle are congruent to the three sides of another triangle, then the triangles are congruent.

Three fixed side lengths can build only one triangle shape — there's no wiggle room. That's why three sides are enough.

Worked Example

In $\triangle ABC$ and $\triangle DEF$: $AB = DE = 5, \quad BC = EF = 7, \quad CA = FD = 9$

All three side pairs match, so by SSS, $\triangle ABC \cong \triangle DEF$.

💡 SSS is the postulate behind why triangles make rigid bracing — a triangular frame can't be deformed without changing a side length.

Side-Angle-Side (SAS)

🔑 SAS Postulate: If two sides and the included angle of one triangle are congruent to two sides and the included angle of another, the triangles are congruent.

The included angle is the angle between the two named sides — its vertex is where the two sides meet.

Worked Example

In $\triangle GHI$ and $\triangle JKL$: $GH = JK = 6, \quad \angle H = \angle K = 40°, \quad HI = KL = 8$

Here $\angle H$ sits between sides $\overline{GH}$ and $\overline{HI}$ — it's included. So by SAS, $\triangle GHI \cong \triangle JKL$.

⚠️ Order matters in SAS: the angle must be between the two sides. "Side-Side-Angle" (a side, another side, then a non-included angle) does not prove congruence — that's the SSA trap we'll meet in Part 4.

Find the Included Angle / Side 🔽

The included angle between two sides is at the vertex they share; the included side between two angles is the segment connecting their vertices.

A Free Side or Angle: the Reflexive Property

Many figures have two triangles that share a side or angle. A shared part is congruent to itself:

🔑 Reflexive Property: $\overline{XY} \cong \overline{XY}$ and $\angle X \cong \angle X$.

Worked Example

In a kite, segment $\overline{BD}$ is shared by $\triangle ABD$ and $\triangle CBD$. If $\overline{AB}\cong\overline{CB}$ and $\overline{AD}\cong\overline{CD}$, then together with the shared side $\overline{BD}\cong\overline{BD}$ we have three pairs of sides — so $\triangle ABD \cong \triangle CBD$ by SSS.

💡 The shared side is the "third pair" you'd otherwise be missing. Spotting reflexive parts is one of the most useful proof skills in this whole unit.

Concept Check 🎯

Reading Lengths from a Congruence

Once SSS or SAS proves two triangles congruent, every corresponding length and angle is equal (CPCTC). That lets you fill in unknown measurements directly — no extra work needed.

💡 If $\triangle GHI \cong \triangle JKL$, then $\overline{GH}\cong\overline{JK}$, $\overline{HI}\cong\overline{KL}$, $\overline{GI}\cong\overline{JL}$, and the matching angles are equal too. Try it below.

Use the Congruence 🧮

Suppose $\triangle GHI \cong \triangle JKL$ with $GH = 6$, $HI = 8$, and $\angle H = 40°$.

1) Find $JK$ (the side matching $\overline{GH}$). 2) Find $\angle K$ (the angle matching $\angle H$), in degrees. 3) If also $GI = 11$, find $JL$.

📐 Congruent Triangles

Part 3 of 5 — ASA, AAS & the HL Shortcut

🔑 The Idea: Sometimes you know more about the angles than the sides. ASA and AAS use two angles plus one side. HL is a special shortcut just for right triangles.

Angle-Side-Angle (ASA)

🔑 ASA Postulate: If two angles and the included side of one triangle are congruent to two angles and the included side of another, the triangles are congruent.

The included side is the side between the two named angles (it joins their vertices).

Worked Example

In $\triangle ABC$ and $\triangle DEF$: $\angle A = \angle D = 50°, \quad AB = DE = 7, \quad \angle B = \angle E = 60°$

Side $\overline{AB}$ lies between $\angle A$ and $\angle B$ — it's included. So by ASA, $\triangle ABC \cong \triangle DEF$.

💡 Once two angles are fixed, the third angle is forced (the three must sum to $180°$). The single known side then pins down the exact size.

The Third Angle Is Forced 🧮

ASA and AAS work because the triangle's angles must sum to $180°$, so two known angles fix the third. Find the missing angle (in degrees).

1) Two angles are $50°$ and $60°$. Find the third. 2) Two angles are $40°$ and $75°$. Find the third. 3) A right triangle has one angle of $32°$. Find the remaining acute angle.

Angle-Angle-Side (AAS)

🔑 AAS Theorem: If two angles and a non-included side of one triangle are congruent to the corresponding parts of another, the triangles are congruent.

The difference from ASA is where the side sits:

AAS works because two angles determine the third, which effectively converts the picture back into an ASA situation.

Worked Example

In $\triangle GHI$ and $\triangle JKL$: $\angle G = \angle J = 40°, \quad \angle H = \angle K = 75°, \quad HI = KL = 9$

Side $\overline{HI}$ is not between $\angle G$ and $\angle H$ (it touches $H$ but not $G$), so this is AAS, giving $\triangle GHI \cong \triangle JKL$.

⚠️ AAA is NOT enough. Three matching angles only guarantee the same shape (similar triangles), not the same size — think of two equilateral triangles, one tiny and one huge.

ASA or AAS? 🔽

Decide which postulate the given information matches.

The HL Theorem (Right Triangles Only)

Right triangles get one extra shortcut.

🔑 HL (Hypotenuse-Leg) Theorem: If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and a leg of another right triangle, the triangles are congruent.

Three conditions must hold to use HL:

1. Both triangles are right triangles (a $90°$ angle).
2. The hypotenuses are congruent.
3. One pair of legs is congruent.

Why It Works

By the Pythagorean Theorem, $\text{leg}_1^2 + \text{leg}_2^2 = \text{hyp}^2$. If the hypotenuse and one leg match, the second leg is forced to match too — which is really SSS in disguise.

💡 HL is the only time a "Side-Side-Angle" arrangement works — because the angle is a right angle and the side opposite it is the longest (the hypotenuse), removing the ambiguity that normally dooms SSA.

Concept Check 🎯

📐 Congruent Triangles

Part 4 of 5 — Choosing a Method, Proofs & the Traps

🔑 The Idea: Five valid shortcuts — SSS, SAS, ASA, AAS, HL — and two famous frauds, SSA and AAA. This part teaches you to pick the right one and write a clean two-column proof.

The Five Valid Shortcuts (and the Two Frauds)

⚠️ Why SSA fails: With two sides and a non-included angle, the third side can sometimes swing to two different positions — building two non-congruent triangles. The lone exception is HL, where the angle is a right angle.

Which Method? 🎯

A Worked Two-Column Proof

Given: $\overline{AB}\cong\overline{CB}$ and $\overline{BD}$ bisects $\angle ABC$. Prove: $\triangle ABD \cong \triangle CBD$.

Notice the order S-A-S: side (1), included angle (3), side (4). The angle in step 3 sits between the two sides — exactly what SAS requires.

💡 Proof strategy: mark what's given, hunt for "free" parts (shared sides/angles = Reflexive, vertical angles, parallel-line angles), count to three matching parts, then name the postulate.

Complete the Proof 🔽

Given: $\overline{AC}$ and $\overline{BD}$ intersect at $E$, with $E$ the midpoint of both. Prove: $\triangle AEB \cong \triangle CED$. Fill in each reason or step.

Using CPCTC After the Proof

Once the triangles are congruent, CPCTC unlocks any remaining part.

Example

Suppose you've proven $\triangle AEB \cong \triangle CED$ above. You can now conclude:

$\overline{AB}\cong\overline{CD} \quad\text{and}\quad \angle A \cong \angle C$

…even though those weren't in the "Given." Each follows because Corresponding Parts of Congruent Triangles are Congruent.

🔑 The two-step pattern for almost every congruence proof:

1. Prove $\triangle_1 \cong \triangle_2$ (using SSS/SAS/ASA/AAS/HL).
2. Conclude the specific side/angle you want by CPCTC.

Concept Check 🎯

📐 Congruent Triangles

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) read a congruence statement, (2) apply SSS, SAS, ASA, AAS, and HL, (3) avoid the SSA and AAA traps, and (4) finish proofs with CPCTC. Time to put it all together.

Quick Reference

⚠️ Never trust SSA (ambiguous) or AAA (right shape, wrong size). And always check for "free" congruent parts: shared sides/angles (Reflexive) and vertical angles.

Numeric Warm-Up 🧮

Use the Triangle Angle-Sum Theorem ($\angle$s of a triangle add to $180°$) and CPCTC.

1) $\triangle ABC \cong \triangle DEF$ with $\angle A = 55°$ and $\angle B = 65°$. Find $\angle F$ (in degrees). (Hint: $\angle F \cong \angle C$.) 2) $\triangle ABC \cong \triangle DEF$ with $DE = 12$. Find $AB$. 3) In a triangle, two angles measure $90°$ and $35°$. Find the third angle (in degrees).

Choosing a Method, Fast

For each problem, ask three questions in order:

1. Right triangle with hypotenuse + leg? → HL.
2. How many sides vs. angles do I know? Three sides → SSS. Two sides + one angle → check if the angle is included (SAS) or not (then it's SSA — reject).
3. Two angles + one side? Is the side between them (ASA) or off to the side (AAS)?

💡 Don't forget the "free" parts — a shared side/angle (Reflexive) or vertical angles often supply the third piece you need.

Mixed Practice 🎯

One-Line Recall

When you spot the given parts, name the shortcut by what kind of parts and their order:

🔑 S/A pattern → postulate

• SSS · SAS (angle included) · ASA (side included) · AAS (side not included) · HL (right triangle)
• Reject SSA and AAA every time.

Run the pattern on the next set.

Name That Postulate 🔽

Pick the shortcut justified by each set of congruent parts.

Exit Quiz ✅

Answer all three to finish the lesson.