Task Sets: Use congruence and similarity to solve - Common Core Math - HS Geometry

1.

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

A.

SSA — two sides and a non-included angle

B.

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

C.

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

D.

SSS — three pairs of congruent sides

2.

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.

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

3.

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 $CH$ .

4.

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 $EC$ .

5.

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

A.

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

B.

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

C.

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

D.

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

Exercises: Apply Similarity and Congruence to Solve Problems

Warm-Up: Review What You Know

Fluency Practice