Congruence and similarity criteria for triangles

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HiSET › Congruence and similarity criteria for triangles

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$\bigtriangleup ABC \sim \bigtriangleup DEF$ with scale factor 5:4, with $\bigtriangleup ABC$ the larger triangle.

Complete the sentence: the area of $\bigtriangleup ABC$ is % greater than that of $\bigtriangleup DEF$ .

(Select the closest whole percent)

Explanation

The ratio of the areas of two similar triangles is equal to the square of their scale factor. The scale factor is equal to $\frac{5}{4}$ , so the ratio of the areas is the square of this, or $\left ( \frac{5}{4} \right ) ^{2} = \frac{25}{16}$ .

This makes the area of larger triangle $\bigtriangleup ABC$ equal to $\frac{25}{16} \times 100 % = 156 \frac{1}{4} %$ of that of smaller triangle $\bigtriangleup DEF$ —or, equivalently, $56 \frac{1}{4} %$ greater.

Two of the angles of a triangle are congruent; the third has measure ten degrees greater than either one of the first two. What is the measure of the third angle?

$66 \frac{2}{3} ^{\circ }$

$63 \frac{1}{3} ^{\circ }$

$53 \frac{1}{3} ^{\circ }$

$56 \frac{2}{3} ^{\circ }$

$73\frac{1}{3}^{\circ }$

Explanation

Let be the measure of the third angle. Since its measure is ten degrees greater than either of the others, then the common measure of the other two is . The sum of the measures of the angles of a triangle is 180 degrees, so

To solve for , first ungroup and collect like terms:

Isolate ; first add 20:

Divide by 3:

$3x \div 3 = 200 \div 3$

Since $200 \div 3 = 66 \textup{ R }2$ ,

$x = 66 \frac{2}{3}$ .

The third angle measures $66 \frac{2}{3} ^{\circ }$ .

Two of the angles of a triangle are congruent; the third has measure ten degrees greater than either one of the first two. What is the measure of the third angle?

$66 \frac{2}{3} ^{\circ }$

$63 \frac{1}{3} ^{\circ }$

$53 \frac{1}{3} ^{\circ }$

$56 \frac{2}{3} ^{\circ }$

$73\frac{1}{3}^{\circ }$

Explanation

To solve for , first ungroup and collect like terms:

Isolate ; first add 20:

Divide by 3:

$3x \div 3 = 200 \div 3$

Since $200 \div 3 = 66 \textup{ R }2$ ,

$x = 66 \frac{2}{3}$ .

The third angle measures $66 \frac{2}{3} ^{\circ }$ .

$\bigtriangleup ABC \sim \bigtriangleup DEF$ with scale factor 5:4, with $\bigtriangleup ABC$ the larger triangle.

Complete the sentence: the area of $\bigtriangleup ABC$ is % greater than that of $\bigtriangleup DEF$ .

(Select the closest whole percent)

Explanation

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ such that $\frac{AB}{DE} = \frac{BC}{EF}$ .

Does sufficient information exist to prove that $\bigtriangleup ABC \sim \bigtriangleup DEF$ , and if so, by what postulate or theorem?

Insufficient information exists to prove that $\bigtriangleup ABC \sim \bigtriangleup DEF$ .

$\bigtriangleup ABC \sim \bigtriangleup DEF$ can be proved by the AA Similarity Postulate.

$\bigtriangleup ABC \sim \bigtriangleup DEF$ can be proved by the SAS Similarity Theorem.

$\bigtriangleup ABC \sim \bigtriangleup DEF$ can be prove by the Isosceles Triangle Theorem.

$\bigtriangleup ABC \sim \bigtriangleup DEF$ can be proved by the SAS Inequality Theorem (Hinge Theorem).

Explanation

We are given that, between the triangles, two pairs of corresponding sides are proportional. Without knowing anything else, the proportionality of two pairs of sides is insufficient to prove that the triangles are similar.

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ such that $\frac{AB}{DE} = \frac{BC}{EF}$ .

Does sufficient information exist to prove that $\bigtriangleup ABC \sim \bigtriangleup DEF$ , and if so, by what postulate or theorem?

Insufficient information exists to prove that $\bigtriangleup ABC \sim \bigtriangleup DEF$ .

$\bigtriangleup ABC \sim \bigtriangleup DEF$ can be proved by the AA Similarity Postulate.

$\bigtriangleup ABC \sim \bigtriangleup DEF$ can be proved by the SAS Similarity Theorem.

$\bigtriangleup ABC \sim \bigtriangleup DEF$ can be prove by the Isosceles Triangle Theorem.

$\bigtriangleup ABC \sim \bigtriangleup DEF$ can be proved by the SAS Inequality Theorem (Hinge Theorem).

Explanation

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ such that $\angle C \cong \angle F$ and $\angle A \cong \angle D$ .

Does sufficient information exist to prove that $\bigtriangleup ABC \sim \bigtriangleup DEF$ , and if so, by what postulate or theorem?

$\bigtriangleup ABC \sim \bigtriangleup DEF$ can be proved by the AA Similarity Postulate.

Insufficient information exists to prove that $\bigtriangleup ABC \sim \bigtriangleup DEF$ .

$\bigtriangleup ABC \sim \bigtriangleup DEF$ can be prove by the SSS Similarity Theorem.

$\bigtriangleup ABC \sim \bigtriangleup DEF$ can be proved by the SAS Similarity Theorem.

$\bigtriangleup ABC \sim \bigtriangleup DEF$ can be prove by the Isosceles Triangle Theorem.

Explanation

We are given that, between the triangles, two pairs of corresponding angles are congruent. By the AA Similarity Postulate, this is enough to prove that $\bigtriangleup ABC \sim \bigtriangleup DEF$ .

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ such that $\frac{AB}{DE} = \frac{BC}{EF}$ and $\angle B \cong \angle E$ .

Does sufficient information exist to prove that $\bigtriangleup ABC \sim \bigtriangleup DEF$ , and if so, by what postulate or theorem?

$\bigtriangleup ABC \sim \bigtriangleup DEF$ can be proved by the SAS Similarity Theorem.

$\bigtriangleup ABC \sim \bigtriangleup DEF$ can be proved by the AA Similarity Postulate.

Insufficient information exists to prove that $\bigtriangleup ABC \sim \bigtriangleup DEF$ .

$\bigtriangleup ABC \sim \bigtriangleup DEF$ can be prove by the SSS Similarity Theorem.

$\bigtriangleup ABC \sim \bigtriangleup DEF$ can be prove by the Isosceles Triangle Theorem.

Explanation

We are given that, between the triangles, two pairs of corresponding sides are proportional, and that a pair of corresponding angles are congruent. The angles that are congruent are the included angles of their respective sides. By the SAS Similarity Postulate, this is enough to prove that $\bigtriangleup ABC \sim \bigtriangleup DEF$ .

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ such that $\frac{AB}{DE} = \frac{BC}{EF}$ and $\angle C \cong \angle F$ .

Does sufficient information exist to prove that $\bigtriangleup ABC \sim \bigtriangleup DEF$ , and if so, by what postulate or theorem?

Insufficient information exists to prove that $\bigtriangleup ABC \sim \bigtriangleup DEF$ .

$\bigtriangleup ABC \sim \bigtriangleup DEF$ can be prove by the SSS Similarity Theorem.

$\bigtriangleup ABC \sim \bigtriangleup DEF$ can be proved by the SAS Similarity Theorem.

$\bigtriangleup ABC \sim \bigtriangleup DEF$ can be proved by the AA Similarity Postulate.

$\bigtriangleup ABC \sim \bigtriangleup DEF$ can be prove by the Isosceles Triangle Theorem.

Explanation

We are given that, between the triangles, two pairs of corresponding sides are proportional, and that a pair of corresponding angles are congruent. If the angles were the included angles of the triangles, then the SAS Similarity Theorem could be applied to prove that $\bigtriangleup ABC \sim \bigtriangleup DEF$ ; however, the two congruent angles are nonincluded, and there is no "SSA" statement that can be applied to prove similarity. Without further information, it cannot be proved that the triangles are similar.

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ such that $\angle C \cong \angle F$ and $\angle A \cong \angle D$ .

Does sufficient information exist to prove that $\bigtriangleup ABC \sim \bigtriangleup DEF$ , and if so, by what postulate or theorem?

$\bigtriangleup ABC \sim \bigtriangleup DEF$ can be proved by the AA Similarity Postulate.

Insufficient information exists to prove that $\bigtriangleup ABC \sim \bigtriangleup DEF$ .

$\bigtriangleup ABC \sim \bigtriangleup DEF$ can be prove by the SSS Similarity Theorem.

$\bigtriangleup ABC \sim \bigtriangleup DEF$ can be proved by the SAS Similarity Theorem.

$\bigtriangleup ABC \sim \bigtriangleup DEF$ can be prove by the Isosceles Triangle Theorem.

Explanation

We are given that, between the triangles, two pairs of corresponding angles are congruent. By the AA Similarity Postulate, this is enough to prove that $\bigtriangleup ABC \sim \bigtriangleup DEF$ .

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