Back to Exercise: Prove theorems using similarity

Exercises: Prove Theorems Using Similarity

Show all work. For proof problems, write complete reasoning in paragraph or two-column form.

Grade 9·21 problems·~35 min·Common Core Math - HS Geometry·standard·hsg-srt-b-4

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Warm-Up: Review What You Know

These problems review skills from previous lessons.

Two triangles share the same vertex angle $A$ . If a second pair of angles is also equal, which similarity criterion guarantees the triangles are similar?

SAS similarity

SSS similarity

AA similarity

HL congruence

Line $DE$ is parallel to line $BC$ , and line $AB$ is a transversal crossing both.
Which statement correctly describes the relationship between $\angle ADE$ and $\angle ABC$ ?

$\angle ADE$ and $\angle ABC$ are supplementary.

$\angle ADE$ and $\angle ABC$ are corresponding angles, so they are equal.

$\angle ADE$ and $\angle ABC$ are alternate interior angles.

$\angle ADE$ and $\angle ABC$ have no relationship unless the lines are perpendicular.

A right triangle has legs of length 5 and 12. What is the length of the hypotenuse?

$\sqrt{119}$

169

Fluency Practice

Apply the Side-Splitter Theorem, its converse, or right-triangle similarity to find unknown values.

$Triangle ABC with parallel segment DE. AD = 6, DB = 4, AE = 9, EC = ?.$

In $\triangle ABC$ , segment $DE \parallel BC$ with $D$ on $\overline{AB}$ and $E$ on $\overline{AC}$ .
Given $AD = 6$ , $DB = 4$ , and $AE = 9$ , find $EC$ .

$Triangle PQR with parallel segment ST. PS = 5, SQ = 3, PT = 7.5, TR = ?.$

In $\triangle PQR$ , segment $ST \parallel QR$ with $S$ on $\overline{PQ}$ and $T$ on $\overline{PR}$ .
Given $PS = 5$ , $SQ = 3$ , and $PT = 7.5$ , find $TR$ .

$Triangle ABC with segment DE. AD = 8, DB = 4, AE = 10, EC = 5. Is DE parallel to BC?$

In $\triangle ABC$ , point $D$ lies on $\overline{AB}$ and point $E$ lies on $\overline{AC}$ .
Given $AD = 8$ , $DB = 4$ , $AE = 10$ , and $EC = 5$ . Is $DE \parallel BC$ ?

Yes — the ratios $AD/DB$ and $AE/EC$ are equal, so by the converse of the Side-Splitter Theorem, $DE \parallel BC$ .

No — the ratios are not equal, so $DE$ is not parallel to $BC$ .

Yes — $DE$ divides two sides of the triangle, so it must always be parallel to the third side.

Cannot be determined without knowing the lengths of $DE$ and $BC$ .

$Right triangle ABC with altitude CD to hypotenuse AB. AD = 4, DB = 9, AC = ?.$

Right triangle $\triangle ABC$ has a right angle at $C$ . The altitude from $C$ meets
hypotenuse $\overline{AB}$ at point $D$ , with $AD = 4$ and $DB = 9$ .
Using the similarity of $\triangle ACD$ and $\triangle ABC$ , find $AC$ .

$Right triangle with altitude CD to hypotenuse. AD = 3, DB = 12, CD = h = ?.$

In right $\triangle ABC$ with right angle at $C$ , altitude $\overline{CD}$ meets
hypotenuse $\overline{AB}$ at $D$ with $AD = 3$ and $DB = 12$ .
Using $\triangle ACD \sim \triangle CBD$ , find the altitude length $CD$ .

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