Geometry Unit 5: Congruent Triangles | Notes and practice

Notes

Angles in triangles

Angle sum: the three angles add to $180^\circ$.
Exterior Angle Theorem: an exterior angle equals the sum of the two remote interior angles.
Third Angles Theorem: two pairs of congruent angles force the third pair to be congruent.

Proving triangles congruent

Method	What you need
SSS	3 pairs of sides
SAS	2 sides and the included angle
ASA	2 angles and the included side
AAS	2 angles and a nonincluded side
HL	Right triangles: hypotenuse and a leg

AAA and SSA do not prove congruence.
Look for shared sides (Reflexive Property) and vertical angles.
The order of letters in $\triangle ABC \cong \triangle DEF$ shows matching parts. After proving congruence, use CPCTC.

Isosceles and equilateral triangles

Congruent sides are opposite congruent angles, and the converse is true.
Equilateral triangles are equiangular, with $60^\circ$ angles.

Topics in this unit

Each topic has its own short notes and practice.

Practice

In $\triangle ABC$, $m\angle A = (x + 15)^\circ$, $m\angle B = (2x)^\circ$ and $m\angle C = (3x - 15)^\circ$. Find $m\angle B$.

Segments $\overline{AE}$ and $\overline{BD}$ bisect each other at $C$. Which postulate proves $\triangle ACB \cong \triangle ECD$?

$\triangle KLM \cong \triangle RST$. Which angle is congruent to $\angle M$?

An exterior angle of an isosceles triangle at the vertex angle measures $100^\circ$. What is each base angle?

Which information is NOT enough to prove $\triangle ABC \cong \triangle DEF$?

An equilateral triangle has sides $3x - 1$, $2x + 4$ and $y + 2$. Find $y$.

Free response

Write your answer first, then open the scoring guide.

Free-response question

A roof truss is shaped like isosceles triangle $ABC$ with $\overline{AB} \cong \overline{CB}$. A vertical support $\overline{BD}$ runs from the peak $B$ to the midpoint $D$ of the base $\overline{AC}$.

(a) Prove that $\triangle ABD \cong \triangle CBD$. Name the postulate.

Show the answer and scoring guide

$\overline{AB} \cong \overline{CB}$ (given), $\overline{AD} \cong \overline{CD}$ (definition of midpoint), $\overline{BD} \cong \overline{BD}$ (Reflexive). So $\triangle ABD \cong \triangle CBD$ by SSS.

Check off the points you earned:

1 point: lists the three pairs of congruent sides with reasons
1 point: names SSS

(b) Explain why the support is perpendicular to the base.

Show the answer and scoring guide

By CPCTC, $\angle ADB \cong \angle CDB$. They form a linear pair, so they add to $180^\circ$, and each is $90^\circ$. So $\overline{BD} \perp \overline{AC}$.

Check off the points you earned:

1 point: uses CPCTC to get congruent angles at $D$
1 point: uses the linear pair to show each is $90^\circ$

(c) The roof's peak angle $\angle ABC$ is $110^\circ$. Find each base angle and $m\angle ABD$.

Show the answer and scoring guide

Base angles: $\displaystyle \frac{180 - 110}{2} = 35^\circ$ each. $\angle ABD \cong \angle CBD$ by CPCTC, so $m\angle ABD = 55^\circ$.

Check off the points you earned:

1 point: base angles $35^\circ$
1 point: $m\angle ABD = 55^\circ$