Exercises: Prove theorems using similarity

A

Warm-Up: Review What You Know

These problems review skills from previous lessons.

1.

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?

A.

SAS similarity

B.

SSS similarity

C.

AA similarity

D.

HL congruence

2.

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 $\angle ABC$ ?

A.

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

B.

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

C.

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

D.

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

3.

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

A.

17

B.

$\sqrt{119}$

C.

13

D.

The three triangles formed by the altitude are always congruent to the original.

B

Fluency Practice

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

1.

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 $EC$ .

2.

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 $TR$ .

3.

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 $DE \parallel BC$ ?

A.

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

B.

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

C.

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

D.

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

4.

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 $AC$ .

5.

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 $CD$ .

C

Varied Practice

These problems present the same theorems in different formats. Read each carefully.

1.

In $\triangle XYZ$ , segment $MN \parallel YZ$ with $M$ on $\overline{XY}$ and $N$ on $\overline{XZ}$ .
The full length $XY = 18$ and $XM = 12$ . The full length $XZ = 15$ . Find $XN$ .

2.

The diagram shows $\triangle ABC$ with segment $DE$ where $D$ is on $\overline{AB}$ and $E$ is on $\overline{AC}$ .
Given $AD = 4$ , $DB = 2$ , $AE = 6$ , $EC = 3$ .
A student computes $AD/DB = 2$ and $AE/EC = 2$ , then writes:
"By the Side-Splitter Theorem, $DE \parallel BC$ ."
Is this reasoning correct?

A.

Correct reasoning — the ratios being equal proves $DE \parallel BC$ by the Side-Splitter Theorem.

B.

Correct conclusion, but wrong theorem name — the ratios being equal proves $DE \parallel BC$
by the converse of the Side-Splitter Theorem, not the direct theorem.

C.

Incorrect — equal ratios only prove similarity, not parallelism.

D.

Incorrect — parallelism requires angle measurements, not ratios.

3.

In right $\triangle ABC$ with right angle at $C$ , altitude $\overline{CD}$ meets $\overline{AB}$ at $D$ .
Given $AD = x$ , $DB = x + 5$ , and $CD = 6$ .
Use the geometric mean relationship $CD^2 = AD \cdot DB$ to find the positive value of $x$ .

4.

A student says: "The Side-Splitter Theorem tells me that any line segment crossing two sides of
a triangle creates proportional pieces."
Identify the flaw in this statement and write a corrected version in one or two sentences.

5.

In right $\triangle ABC$ with right angle at $C$ , altitude $\overline{CD}$ is drawn to
hypotenuse $\overline{AB}$ . Which similarity statement is correct?

A.

$\triangle ABC \sim \triangle ACD \sim \triangle CBD$ — all three triangles are similar to each other.

B.

$\triangle ACD \cong \triangle CBD$ — the two smaller triangles are congruent.

C.

$\triangle ABC \sim \triangle ACD$ only — $\triangle CBD$ is not similar to the others.

D.

$\triangle ACD$ and $\triangle CBD$ are similar to each other but not to $\triangle ABC$ .

D

Word Problems

Set up an equation using the appropriate theorem, then solve.

1.

A city planning map shows two parallel streets, $\overline{DE}$ and $\overline{BC}$ , cutting
across two roads that meet at vertex $A$ forming $\triangle ABC$ . On the left road, the
block from $A$ to $D$ is 300 m and from $D$ to $B$ is 200 m. On the right road, the block
from $A$ to $E$ is 270 m.

How long is the segment from $E$ to $C$ on the right road?

2.

A carpenter installs a diagonal cross-brace $\overline{DE}$ inside a triangular roof frame
$\triangle ABC$ . The brace runs parallel to the base $\overline{BC}$ . The left rafter
$\overline{AB}$ is 15 ft from peak $A$ to the brace at $D$ , then 10 ft from $D$ down to $B$ .
The right rafter $\overline{AC}$ is 12 ft from peak $A$ to the brace at $E$ .

1.

Find $EC$ , the length of the right rafter below the brace.

2.

What fraction of the total right-rafter length $AC$ does the segment $AE$ represent?
Express as a simplified fraction.

3.

A student is assigned this theorem to prove: "In any right triangle, the altitude drawn from
the right-angle vertex to the hypotenuse creates two smaller triangles, each similar to the
original triangle and to each other."

Outline a proof of this theorem. Identify the similarity criterion used and the pairs of
equal angles for each similarity. Write 3–5 sentences or bullet points.

E

Error Analysis

Each problem shows a student's work that contains an error. Identify the mistake.

1.

Jordan is solving: "In $\triangle ABC$ , $D$ is on $\overline{AB}$ and $E$ is on $\overline{AC}$
with $AD = 6$ , $DB = 4$ , $AE = 9$ , $EC = 6$ . Is $DE$ parallel to $BC$ ?"

Jordan writes:

$AD/DB = 6/4 = 1.5$
$AE/EC = 9/6 = 1.5$
The ratios are equal, so by the Side-Splitter Theorem, $DE \parallel BC$ .

Jordan reaches the right conclusion but cites the wrong theorem. Which theorem should be cited,
and why?

A.

Jordan should cite the converse of the Side-Splitter Theorem — the direct theorem goes
the other direction (parallel → proportional), not the direction Jordan used.

B.

Jordan should cite the AA similarity criterion, because proportional ratios imply similar triangles.

C.

Jordan's citation is correct — the Side-Splitter Theorem applies whenever ratios are equal.

D.

Jordan should cite the Pythagorean Theorem, which governs all triangle side relationships.

2.

In right $\triangle ABC$ with right angle at $C$ and altitude $\overline{CD}$ to hypotenuse,
Priya writes the similarity $\triangle ABC \sim \triangle ACD$ and sets up:
$\frac{AB}{AC} = \frac{AC}{CD}$
claiming that $AB$ corresponds to $AC$ , and $AC$ corresponds to $CD$ .

Priya's proportion is incorrect. Which error did she make?

A.

She used the wrong similarity criterion — she should use SSS instead of AA.

B.

She misidentified corresponding sides. From $\triangle ABC \sim \triangle ACD$ , the correct
proportion is $\frac{AB}{AC} = \frac{BC}{CD} = \frac{AC}{AD}$ , not $\frac{AB}{AC} = \frac{AC}{CD}$ .

C.

She should not use $\triangle ACD$ because the altitude may fall outside the triangle.

D.

The similarity $\triangle ABC \sim \triangle ACD$ is itself wrong — only the two sub-triangles
are similar.

F

Challenge / Extension

Bonus problems for extra practice. Show complete reasoning.

1.

Prove the Triangle Midsegment Theorem using the converse of the Side-Splitter Theorem:
if $D$ and $E$ are the midpoints of $\overline{AB}$ and $\overline{AC}$ in $\triangle ABC$ ,
then $DE \parallel BC$ .
Write a complete proof stating what is given, what the theorem says, and how it applies.