QuestionWhich statement is true for triangles $A B C$ and $D E F$ with $A B=9, B C=15, D E=6, E F=10$ , and $\angle B \cong \angle E$ ? a. $\angle C A B \cong \angle D E F$ b. $\frac{A B}{C B}=\frac{F E}{D E}$ c. $\triangle A B C \cong \triangle D E F$ d. $\frac{A B}{D E}=\frac{F E}{C B}$

Studdy Solution

STEP 1

Assumptions1. Triangle ABC has sides AB =9, BC =15 and angle B. . Triangle DEF has sides DE =6, EF =10 and angle.
3. Angle B is congruent to angle.

STEP 2

Since angle B is congruent to angle, triangles ABC and DEF are similar by the Angle-Angle (AA) similarity postulate. This postulate states that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

STEP 3

When two triangles are similar, the ratios of the lengths of their corresponding sides are equal. So, we can write the ratios of the corresponding sides of triangles ABC and DEF as follows $\frac{AB}{DE} = \frac{BC}{EF}$

STEP 4

Substitute the given values into the equation $\frac{9}{6} = \frac{15}{10}$

STEP 5

implify both sides of the equation $\frac{3}{2} = \frac{3}{2}$ This equation is true, so the ratios of the lengths of the corresponding sides of triangles ABC and DEF are indeed equal.

STEP 6

Now, let's check each statement given in the problem to see which one is true.
a. $\angle CAB \cong \angle DEF$
We don't have enough information to determine whether this statement is true, so we can't choose this option.
b. $\frac{AB}{CB} = \frac{FE}{DE}$ This is not the correct ratio of the corresponding sides of the similar triangles, so this statement is false.
c. $\triangle ABC \cong \triangle DEF$
The triangles are similar, not congruent, so this statement is false.
d. $\frac{AB}{DE} = \frac{FE}{CB}$ This is the correct ratio of the corresponding sides of the similar triangles, so this statement is true.
The correct answer is d. $\frac{AB}{DE} = \frac{FE}{CB}$ .

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QuestionWhich statement is true for triangles $A B C$ and $D E F$ with $A B=9, B C=15, D E=6, E F=10$ , and $\angle B \cong \angle E$ ? a. $\angle C A B \cong \angle D E F$ b. $\frac{A B}{C B}=\frac{F E}{D E}$ c. $\triangle A B C \cong \triangle D E F$ d. $\frac{A B}{D E}=\frac{F E}{C B}$

Studdy Solution

STEP 1

Assumptions1. Triangle ABC has sides AB =9, BC =15 and angle B. . Triangle DEF has sides DE =6, EF =10 and angle.
3. Angle B is congruent to angle.

STEP 2

Since angle B is congruent to angle, triangles ABC and DEF are similar by the Angle-Angle (AA) similarity postulate. This postulate states that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

STEP 3

When two triangles are similar, the ratios of the lengths of their corresponding sides are equal. So, we can write the ratios of the corresponding sides of triangles ABC and DEF as follows $\frac{AB}{DE} = \frac{BC}{EF}$

STEP 4

Substitute the given values into the equation $\frac{9}{6} = \frac{15}{10}$

STEP 5

implify both sides of the equation $\frac{3}{2} = \frac{3}{2}$ This equation is true, so the ratios of the lengths of the corresponding sides of triangles ABC and DEF are indeed equal.

STEP 6

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Studdy Solution

STEP 1

Assumptions1. Triangle ABC has sides AB =9, BC =15 and angle B. . Triangle DEF has sides DE =6, EF =10 and angle.3. Angle B is congruent to angle.

STEP 2

Since angle B is congruent to angle, triangles ABC and DEF are similar by the Angle-Angle (AA) similarity postulate. This postulate states that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

STEP 3

When two triangles are similar, the ratios of the lengths of their corresponding sides are equal. So, we can write the ratios of the corresponding sides of triangles ABC and DEF as followsABDE=BCEF\frac{AB}{DE} = \frac{BC}{EF}DEAB​=EFBC​

STEP 4

Substitute the given values into the equation96=1510\frac{9}{6} = \frac{15}{10}69​=1015​

STEP 5

implify both sides of the equation32=32\frac{3}{2} = \frac{3}{2}23​=23​This equation is true, so the ratios of the lengths of the corresponding sides of triangles ABC and DEF are indeed equal.

STEP 6

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Assumptions1. Triangle ABC has sides AB =9, BC =15 and angle B. . Triangle DEF has sides DE =6, EF =10 and angle.
3. Angle B is congruent to angle.

When two triangles are similar, the ratios of the lengths of their corresponding sides are equal. So, we can write the ratios of the corresponding sides of triangles ABC and DEF as follows $\frac{AB}{DE} = \frac{BC}{EF}$

Substitute the given values into the equation $\frac{9}{6} = \frac{15}{10}$

implify both sides of the equation $\frac{3}{2} = \frac{3}{2}$ This equation is true, so the ratios of the lengths of the corresponding sides of triangles ABC and DEF are indeed equal.