Back to Exercise: Use congruence and similarity to solve

Exercises: Apply Similarity and Congruence to Solve Problems

Work through each section in order. For proof problems, state the criterion used (SSS, SAS, ASA, AAS, AA, SAS~, SSS~) and identify CPCTC steps explicitly. Show all work for computation problems.

Grade 9·22 problems·~30 min·Common Core Math - HS Geometry·standard·hsg-srt-b-5

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Warm-Up: Review What You Know

These problems review skills you already know.

Two triangles share a side. You know two angles of the first triangle are congruent to two angles of the second triangle. Which congruence criterion applies?

SSS — three pairs of congruent sides

AAS — two angles and a non-included side are congruent

SAS — two sides and the included angle are congruent

The triangles are congruent because they look the same size

Triangle $PQR$ is similar to triangle $XYZ$ with a scale factor of 3. If $PQ = 6$ , what is $XY$ ?

6 — similar triangles are the same size

$Triangle ABC with segment DE parallel to BC, creating smaller triangle ADE inside triangle ABC.$

In the diagram, $DE \parallel BC$ with $D$ on $\overline{AB}$ and $E$ on $\overline{AC}$ . Which similarity criterion proves $\triangle ADE \sim \triangle ABC$ ?

SSS — three pairs of proportional sides

SAS~ — two proportional sides and the included angle

AA — two pairs of congruent angles

SSA — two sides and a non-included angle

Fluency Practice

Apply congruence or similarity criteria directly. State the criterion used.

In parallelogram $ABCD$ , diagonal $\overline{AC}$ is drawn. You know $AB \parallel CD$ and $BC \parallel AD$ . Which criterion proves $\triangle ABC \cong \triangle CDA$ ?

SSA — two sides and a non-included angle

SAS — $AC = CA$ , $\angle BAC = \angle DCA$ , $\angle BCA = \angle DAC$

ASA — $\angle BAC \cong \angle DCA$ , $AC \cong CA$ , $\angle BCA \cong \angle DAC$

SSS — three pairs of congruent sides

In rectangle $ABCD$ , $\overline{AB} \cong \overline{DC}$ , $\overline{BC} \cong \overline{BC}$ (reflexive), and $\angle ABC \cong \angle DCB$ (both right angles). A student concludes: "By CPCTC, $\overline{AC} \cong \overline{DB}$ ." Identify the missing step and state the complete proof.

$Right triangle ABC with altitude CH to hypotenuse AB. AH = 4, HB = 9, CH = ?$

In right triangle $ABC$ with the right angle at $C$ , the altitude from $C$ to hypotenuse $\overline{AB}$ meets $\overline{AB}$ at point $H$ . If $AH = 4$ and $HB = 9$ , find the length of altitude $CH$ .

In $\triangle ABC$ , point $D$ is on $\overline{AB}$ and point $E$ is on $\overline{AC}$ such that $DE \parallel BC$ . Given $AD = 8$ , $DB = 4$ , and $AE = 12$ , find $EC$ .

In isosceles triangle $PQR$ with $PQ \cong PR$ , $M$ is the midpoint of $\overline{QR}$ . Which congruence criterion proves $\triangle PQM \cong \triangle PRM$ ?

AAS — $\angle PQM \cong \angle PRM$ , $\angle QPM \cong \angle RPM$ , $QM \cong RM$

SAS — $PQ \cong PR$ , $\angle QPM \cong \angle RPM$ , $PM \cong PM$

SSS — $PQ \cong PR$ , $QM \cong RM$ , $PM \cong PM$

ASA — $\angle QPM \cong \angle RPM$ , $PM \cong PM$ , $\angle PMQ \cong \angle PMR$

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