Back to Exercise: Prove theorems about triangles

Exercises: Prove Theorems About Triangles

Work through each section in order. Show your reasoning for proof-based problems.

Grade 9·21 problems·~35 min·Common Core Math - HS Geometry·standard·hsg-co-c-10

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Warm-Up: Review What You Know

These problems review skills you already have.

Lines $\ell$ and $m$ are parallel, cut by a transversal $t$ . Which theorem guarantees that the alternate interior angles are congruent?

Vertical Angles Theorem

Alternate Interior Angles Theorem

Corresponding Angles Postulate

Same-Side Interior Angles Theorem

Triangle $ABC$ has $AB = AC$ . An angle bisector is drawn from $A$ to point $D$ on $\overline{BC}$ . Which congruence criterion proves $\triangle ABD \cong \triangle ACD$ ?

SSS (Side-Side-Side)

ASA (Angle-Side-Angle)

SAS (Side-Angle-Side)

AAS (Angle-Angle-Side)

Point $D$ is the midpoint of $\overline{AB}$ , where $A = (0, 0)$ and $B = (8, 0)$ . What are the coordinates of $D$ ?

$(4, 0)$

$(2, 0)$

$(8, 0)$

$(0, 4)$

Fluency Practice

Apply the triangle theorems directly to find missing values.

$Triangle PQR with angles 47° at P, 68° at Q, and an unknown angle at R.$

In triangle $PQR$ , $m\angle P = 47^\circ$ and $m\angle Q = 68^\circ$ . Find $m\angle R$ .

Which theorem from HSG.CO.C.9 is the essential step in the proof of the Triangle Angle Sum Theorem?

Vertical Angles Theorem — vertical angles at a point sum to 180°

Alternate Interior Angles Theorem — when a transversal cuts parallel lines, alternate interior angles are congruent

Exterior Angle Theorem — an exterior angle equals the sum of the two remote interior angles

Corresponding Angles Postulate — corresponding angles formed by a transversal are supplementary

$Isosceles triangle ABC with vertex angle 50° at A and equal legs AB = AC. The base angle at B is unknown.$

Isosceles triangle $ABC$ has $AB = AC$ . The vertex angle measures $\angle A = 50^\circ$ . Find $m\angle B$ .

In equilateral triangle $XYZ$ , all three sides are equal. What is $m\angle X$ ?

In triangle $ABC$ , $D$ is the midpoint of $\overline{AB}$ and $E$ is the midpoint of $\overline{AC}$ . If $BC = 24$ , find the length of midsegment $\overline{DE}$ .

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