Exercises: Explain triangle congruence criteria - Common Core Math - HS Geometry

A

Warm-Up: Review What You Know

These problems review prerequisite skills from CO.B.6, CO.B.7, and Grade 8 geometry.

1.

According to the definition from HSG.CO.B.7, two triangles are congruent if and only if:

A.

All three pairs of corresponding angles are congruent.

B.

There exists a sequence of rigid motions mapping one triangle onto the other.

C.

All three pairs of corresponding sides are congruent.

D.

The triangles have the same area.

2.

CPCTC stands for "Corresponding Parts of Congruent Triangles are Congruent." In a geometric proof, CPCTC is used:

A.

As a reason to establish that two triangles are congruent.

B.

After congruence has been established, to conclude that specific pairs of sides or angles are equal.

C.

To prove that a triangle is isosceles.

D.

As a substitute for identifying which congruence criterion applies.

3.

In triangle $\triangle PQR$ , $\angle P = 48^\circ$ and $\angle Q = 75^\circ$ . What is the measure of $\angle R$ ?

A.

$57^\circ$

B.

$123^\circ$

C.

$67^\circ$

D.

$48^\circ$

B

Fluency Practice

1.

Two triangles have the following known congruent parts: $\overline{AB} \cong \overline{DE}$ , $\angle A \cong \angle D$ , and $\overline{AC} \cong \overline{DF}$ . The angle $\angle A$ is between sides $\overline{AB}$ and $\overline{AC}$ . Which criterion guarantees the triangles are congruent?

A.

SSS

B.

ASA

C.

SAS

D.

SSA — two sides and a non-included angle

2.

In the SAS rigid-motion proof, after translating vertex $A$ to vertex $D$ and rotating so that $B$ maps to $E$ , what forces vertex $C''$ to land exactly on vertex $F$ ?

A.

The included angle $\angle A \cong \angle D$ forces ray $DC''$ to point in the same direction as ray $DF$ , and the side length $AC = DF$ places $C''$ at the correct distance.

B.

The triangle angle sum forces $\angle C'' = \angle F$ , which places $C''$ on $F$ .

C.

Two circles centered at $D$ and $E$ intersect at $F$ , determining $C''$ .

D.

A reflection over line $DE$ maps $C''$ to $F$ .

3.

In the ASA proof, after aligning side $\overline{AB}$ onto $\overline{DE}$ (so $A'' = D$ and $B'' = E$ ), why is vertex $C''$ uniquely determined as $F$ ?

A.

The side $AC = DF$ places $C''$ at the correct distance from $D$ .

B.

The angle at $A$ forces ray $DC''$ onto ray $DF$ , and the angle at $B$ forces ray $EC''$ onto ray $EF$ ; their intersection is uniquely $F$ .

C.

Two circles centered at $D$ and $E$ with radii $DF$ and $EF$ intersect at $F$ .

D.

The third angle $\angle C = \angle F$ is automatically equal, forcing the vertex to coincide.

4.

In $\triangle ABC$ and $\triangle DEF$ , you know $\angle A \cong \angle D$ , $\angle C \cong \angle F$ , and $\overline{BC} \cong \overline{EF}$ . This is AAS (two angles and a non-included side). Which statement correctly explains why these triangles must be congruent?

A.

AAS is a separate valid criterion, proved independently of ASA.

B.

AAS does not guarantee congruence because the side is not included between the two angles.

C.

Since $\angle A + \angle C + \angle B = 180^\circ$ and $\angle D + \angle F + \angle E = 180^\circ$ , we get $\angle B \cong \angle E$ ; now $\angle B \cong \angle E$ , $\overline{BC} \cong \overline{EF}$ , and $\angle C \cong \angle F$ is ASA applied to side $\overline{BC}$ .

D.

AAS fails like SSA because neither has the side between the two known angles.

5.

In the SSS rigid-motion proof for $\triangle ABC \cong \triangle DEF$ (with $AB = DE$ , $AC = DF$ , $BC = EF$ ), after translating $A$ to $D$ and rotating so $B$ maps to $E$ , vertex $C''$ must satisfy $DC'' = DF$ and $EC'' = EF$ . Why does this guarantee that $C''$ is either $F$ or the reflection of $F$ over line $DE$ ?

A.

Because the angle sum forces $\angle DC''E = \angle DFE$ .

B.

Because $C''$ lies on two circles — one centered at $D$ with radius $DF$ , one centered at $E$ with radius $EF$ — and two distinct circles intersect in at most two points, which are symmetric about the line through their centers.

C.

Because SAS applied to the two sub-triangles forces $C'' = F$ .

D.

Because a translation followed by a rotation always produces a unique image.

C

Mixed Practice

D

Word Problems

1.

In the figure, point $M$ is the midpoint of both $\overline{AC}$ and $\overline{BD}$ . This means $\overline{AM} \cong \overline{CM}$ and $\overline{BM} \cong \overline{DM}$ .

1.

Name the congruence criterion that proves $\triangle AMB \cong \triangle CMD$ , and list the three pairs of congruent parts that justify it.

2.

Using the congruence from part (a) and CPCTC, what can you conclude about $\overline{AB}$ and $\overline{CD}$ ?

2.

A surveyor needs to find the distance across a river from point $A$ on one bank to point $B$ on the opposite bank. She marks a point $C$ on her side of the river such that $\angle ACB = 90^\circ$ . She then walks along the bank to point $D$ so that $C$ is the midpoint of $\overline{AD}$ . Finally, she walks inland to point $E$ on the opposite bank so that $D$ , $C$ , and $B$ are collinear, $\angle EDC = 90^\circ$ , and $D$ , $E$ , $C$ , $B$ are configured so that $\triangle EDC \cong \triangle BAC$ . Identify which congruence criterion proves $\triangle BAC \cong \triangle EDC$ and name the pair of sides that tells her the river width $AB$ .

3.

In isosceles triangle $\triangle PQR$ , $\overline{PQ} \cong \overline{PR}$ . Let $M$ be the midpoint of $\overline{QR}$ . Draw segment $\overline{PM}$ . Use SSS to prove that $\triangle PQM \cong \triangle PRM$ , and then use CPCTC to explain why $\overline{PM}$ is the perpendicular bisector of $\overline{QR}$ .

E

Error Analysis

1.

A student wrote the following proof:

Given: In $\triangle ABC$ and $\triangle DEF$ : $AB = DE = 8$ , $BC = EF = 5$ , $\angle A \cong \angle D$ ( $\angle A$ is opposite side $BC$ and $\angle D$ is opposite side $EF$ ).

Student's conclusion: "I have two sides and an angle equal in both triangles, so by SAS, $\triangle ABC \cong \triangle DEF$ ."

The student's conclusion is incorrect. Which statement best identifies and explains the error?

A.

The student used the wrong side lengths — SAS requires all three sides to match.

B.

The student confused SAS with SSA. The angle $\angle A$ is opposite side $BC$ , not between sides $AB$ and $BC$ , so this is SSA — which does not guarantee congruence.

C.

The student should have used ASA instead of SAS, since two angles are implied.

D.

SAS is valid here because the student correctly identified two sides and an angle.

2.

A student wrote:

"Triangle $\triangle ABC$ has angles $60^\circ$ , $60^\circ$ , $60^\circ$ . Triangle $\triangle DEF$ also has angles $60^\circ$ , $60^\circ$ , $60^\circ$ . All three pairs of corresponding angles are equal, so by AAA, $\triangle ABC \cong \triangle DEF$ ."

What is the fundamental error in the student's reasoning?

A.

The student should have used ASA instead of AAA, since only two of the angles need to be stated.

B.

The student is correct — three equal angle pairs guarantee congruence just as three equal side pairs do.

C.

AAA guarantees similarity (same shape), not congruence (same shape and size). Both triangles have $60^\circ$ angles, but their side lengths could differ — for example, one equilateral triangle with side 3 and another with side 5. A rigid motion preserves distances, so no rigid motion can map one to the other.

D.

The student needs to check a fourth part (the area) to complete the proof.

F

Challenge

1.

The Hypotenuse-Leg (HL) criterion states: if two right triangles have equal hypotenuses and one pair of equal legs, then they are congruent. Explain why HL is valid by showing it reduces to SSS. (Hint: use the Pythagorean theorem to find the third side.)

2.