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HiSET: Math : Congruence and similarity criteria for triangles

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Example Questions

Example Question #1 : Measurement And Geometry

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?

Possible Answers:

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

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

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

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

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

Correct answer:

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

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 x- 10 . The sum of the measures of the angles of a triangle is 180 degrees, so

x+ (x-10) + (x-10) = 180

To solve for , first ungroup and collect like terms:

x+ x - 10 + x - 10 = 180

x+ x + x - 10 - 10 = 180

3x - 20 = 180

Isolate ; first add 20:

3x-20+20 = 180 + 20

3x= 200

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

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Example Question #2 : Measurement And Geometry

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?

Possible Answers:

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

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

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

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

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

Correct answer:

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

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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Example Question #1 : Congruence And Similarity Criteria For Triangles

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?

Possible Answers:

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

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

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

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

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

Correct answer:

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

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.

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Example Question #1 : Measurement And Geometry

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?

Possible Answers:

$\bigtriangleup ABC \sim \bigtriangleup DEF$ 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 Isosceles Triangle Theorem.

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

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

Correct answer:

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

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Example Question #1 : Measurement And Geometry

$\bigtriangleup ABC \sim \bigtriangleup DEF$ 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)

Possible Answers:

Correct answer:

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.

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Example Question #1 : Congruence And Similarity Criteria For Triangles

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?

Possible Answers:

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

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

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

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

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

Correct answer:

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

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.

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Example Question #1 : Measurement And Geometry

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

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

(Select the closest whole percent).

Possible Answers:

Correct answer:

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 $\frac{4} {5}$ , so the ratio of the areas is the square of this, or $\left ( \frac{4}{5} \right ) ^{2} = \frac{16}{25}$ .

This makes the area of smaller triangle $\bigtriangleup ABC$ equal to $\frac{16}{25} \times 100 \% = 64 \%$ of that of larger triangle $\bigtriangleup DEF$ —or, equivalently, $(100 - 64) \% = 36 \%$ less.

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Example Question #3 : Prove Relationships In Geometric Figures

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

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

(Select the closest whole percent).

Possible Answers:

Correct answer:

Explanation:

The ratio of the perimeters of two similar triangles is equal to their scale factor. This is factor is $\frac{5}{4}$ .

This makes the perimeter of larger triangle $\bigtriangleup ABC$ equal to $\frac{5}{4} \times 100 \% = 125 \%$ of that of smaller triangle $\bigtriangleup DEF$ —or, equivalently, 25% greater.

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Example Question #2 : Measurement And Geometry

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

Complete the sentence: the perimeter of $\bigtriangleup ABC$ is _______% less than that of $\bigtriangleup DEF$ .

(Select the closest whole percent)

Possible Answers:

Correct answer:

Explanation:

The ratio of the perimeters of two similar triangles is equal to their scale factor. This is factor is $\frac{4}{5}$ .

This makes the perimeter of the smaller triangle $\bigtriangleup ABC$ equal to $\frac{4}{5} \times 100 \% = 80 \%$ of that of larger triangle $\bigtriangleup DEF$ —or, equivalently, $(100 - 80) \% = 20 \%$ less.

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Example Question #1 : Measurement And Geometry

A triangle has sides of length 8 and 12; the triangle is scalene and obtuse. Which of the following could be the length of its third side?

Possible Answers:

Correct answer:

Explanation:

A scalene triangle has three sides of different lengths. The triangle is known to have sides of length 8 and 12, so this eliminates 8 and 12 as the correct choices for the length of the third side.

The sum of the lengths of the two smallest sides must exceed the length of the third side. 4 can be eliminated as a the correct choice, since 4+ 8 = 12 , violating this condition.

This leaves 6 and 10 as possible answers. For a triangle to be obtuse, it must hold that if a, b, c are its sidelengths, the greatest of the three,

$a^{2}+b^{2} < c^{2}$ .

If the length of the third side is 10, setting a= 10, b= 8, c=12 , we see that

$10^{2}+ 8^{2}= 100 + 64 = 164 > 144 = 12^{2}$ ,

violating this condition.

If the length of the third side is 6, setting a= 6, b= 8, c=12 , we see that

$6^{2}+ 8^{2}= 36 + 64 = 100 < 144 = 12^{2}$ ,

satisfying this condition.

This makes 6 the correct choice.

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HiSET: Math : Congruence and similarity criteria for triangles

Study concepts, example questions & explanations for HiSET: Math

All HiSET: Math Resources

Example Questions

Example Question #1 : Measurement And Geometry

Example Question #2 : Measurement And Geometry

Example Question #1 : Congruence And Similarity Criteria For Triangles

Example Question #1 : Measurement And Geometry

Example Question #1 : Measurement And Geometry

Example Question #1 : Congruence And Similarity Criteria For Triangles

Example Question #1 : Measurement And Geometry

Example Question #3 : Prove Relationships In Geometric Figures

Example Question #2 : Measurement And Geometry

Example Question #1 : Measurement And Geometry

All HiSET: Math Resources

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