How to find if of acute / obtuse isosceles triangle are similar - ACT Math

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Question

Triangle A and Triangle B are similar isosceles triangles. Triangle A's sides measure , , and . Two of the angles in Triangle A each measure $\small 83^o$ . Triangle B's sides measure , , and . What is the measure of the smallest angle in Triangle B?

Answer

Because the interior angles of a triangle add up to $\small 180^o$ , and two of Triangle A's interior angles measure $\small 83^o$ , we must simply add the two given angles and subtract from $\small 180^o$ to find the missing angle:

$\small 83^o+83^o=166^o$

$\small 180^o-166^o=14^o$

Therefore, the missing angle (and the smallest) from Triangle A measures $\small 14^o$ . If the two triangles are similar, their interior angles must be congruent, meaning that the smallest angle is Triangle B is also $\small 14^o$ .

The side measurements presented in the question are not needed to find the answer!

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Question

Triangle A and Triangle B are similar isosceles triangles. Triangle A has a base of and a height of . Triangle B has a base of . What is the length of Triangle B's two congruent sides?

Answer

We must first find the length of the congruent sides in Triangle A. We do this by setting up a right triangle with the base and the height, and using the Pythagorean Theorem to solve for the missing side ( $\small c$ ). Because the height line cuts the base in half, however, we must use for the length of the base's side in the equation instead of . This is illustrated in the figure below:

Using the base of $\small 3cm$ and the height of $\small 4cm$ , we use the Pythagorean Theorem to solve for $\small c$ :

$\small 3^2+4^2=c^2$

$\small 9+16=c^2$

$\small 25=c^2$

$\small c=5$

Therefore, the two congruent sides of Triangle A measure $\small 5cm$ ; however, the question asks for the two congruent sides of Triangle B. In similar triangles, the ratio of the corresponding sides must be equal. We know that the base of Triangle A is and the base of Triangle B is . We then set up a cross-multiplication using the ratio of the two bases and the ratio of $\small c$ to the side we're trying to find ( $\small x$ ), as follows:

$\small \frac{6}{12}=\frac{5}{x}$

$\small 6x=60$

$\small x=10$

Therefore, the length of the congruent sides of Triangle B is .

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Question

Isosceles triangles $\Delta ABC$ and $\Delta ABD$ share common side . $\Delta ABC$ is an obtuse triangle with sides . $\Delta ABD$ is also an obtuse isosceles triangle, where . What is the measure of $\angle AD$ ?

Answer

In order to prove triangle congruence, the triangles must have SAS, SSS, AAS, or ASA congruence. Here, we have one common side (S), and no other demonstrated congruence. Hence, we cannot guarantee that side is not one of the two congruent sides of $\Delta ABD$ , so we cannot state congruence with $\Delta ABC$ .

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Question

Triangle A and Triangle B are similar isosceles triangles. Triangle A's sides measure , , and . Two of the angles in Triangle A each measure $\small 83^o$ . Triangle B's sides measure , , and . What is the measure of the smallest angle in Triangle B?

Answer

Because the interior angles of a triangle add up to $\small 180^o$ , and two of Triangle A's interior angles measure $\small 83^o$ , we must simply add the two given angles and subtract from $\small 180^o$ to find the missing angle:

$\small 83^o+83^o=166^o$

$\small 180^o-166^o=14^o$

Therefore, the missing angle (and the smallest) from Triangle A measures $\small 14^o$ . If the two triangles are similar, their interior angles must be congruent, meaning that the smallest angle is Triangle B is also $\small 14^o$ .

The side measurements presented in the question are not needed to find the answer!

Compare your answer with the correct one above

Question

Triangle A and Triangle B are similar isosceles triangles. Triangle A has a base of and a height of . Triangle B has a base of . What is the length of Triangle B's two congruent sides?

Answer

We must first find the length of the congruent sides in Triangle A. We do this by setting up a right triangle with the base and the height, and using the Pythagorean Theorem to solve for the missing side ( $\small c$ ). Because the height line cuts the base in half, however, we must use for the length of the base's side in the equation instead of . This is illustrated in the figure below:

Using the base of $\small 3cm$ and the height of $\small 4cm$ , we use the Pythagorean Theorem to solve for $\small c$ :

$\small 3^2+4^2=c^2$

$\small 9+16=c^2$

$\small 25=c^2$

$\small c=5$

Therefore, the two congruent sides of Triangle A measure $\small 5cm$ ; however, the question asks for the two congruent sides of Triangle B. In similar triangles, the ratio of the corresponding sides must be equal. We know that the base of Triangle A is and the base of Triangle B is . We then set up a cross-multiplication using the ratio of the two bases and the ratio of $\small c$ to the side we're trying to find ( $\small x$ ), as follows:

$\small \frac{6}{12}=\frac{5}{x}$

$\small 6x=60$

$\small x=10$

Therefore, the length of the congruent sides of Triangle B is .

Compare your answer with the correct one above

Question

Isosceles triangles $\Delta ABC$ and $\Delta ABD$ share common side . $\Delta ABC$ is an obtuse triangle with sides . $\Delta ABD$ is also an obtuse isosceles triangle, where . What is the measure of $\angle AD$ ?

Answer

In order to prove triangle congruence, the triangles must have SAS, SSS, AAS, or ASA congruence. Here, we have one common side (S), and no other demonstrated congruence. Hence, we cannot guarantee that side is not one of the two congruent sides of $\Delta ABD$ , so we cannot state congruence with $\Delta ABC$ .

Compare your answer with the correct one above

Question

Triangle A and Triangle B are similar isosceles triangles. Triangle A's sides measure , , and . Two of the angles in Triangle A each measure $\small 83^o$ . Triangle B's sides measure , , and . What is the measure of the smallest angle in Triangle B?

Answer

Because the interior angles of a triangle add up to $\small 180^o$ , and two of Triangle A's interior angles measure $\small 83^o$ , we must simply add the two given angles and subtract from $\small 180^o$ to find the missing angle:

$\small 83^o+83^o=166^o$

$\small 180^o-166^o=14^o$

Therefore, the missing angle (and the smallest) from Triangle A measures $\small 14^o$ . If the two triangles are similar, their interior angles must be congruent, meaning that the smallest angle is Triangle B is also $\small 14^o$ .

The side measurements presented in the question are not needed to find the answer!

Compare your answer with the correct one above

Question

Triangle A and Triangle B are similar isosceles triangles. Triangle A has a base of and a height of . Triangle B has a base of . What is the length of Triangle B's two congruent sides?

Answer

We must first find the length of the congruent sides in Triangle A. We do this by setting up a right triangle with the base and the height, and using the Pythagorean Theorem to solve for the missing side ( $\small c$ ). Because the height line cuts the base in half, however, we must use for the length of the base's side in the equation instead of . This is illustrated in the figure below:

Using the base of $\small 3cm$ and the height of $\small 4cm$ , we use the Pythagorean Theorem to solve for $\small c$ :

$\small 3^2+4^2=c^2$

$\small 9+16=c^2$

$\small 25=c^2$

$\small c=5$

Therefore, the two congruent sides of Triangle A measure $\small 5cm$ ; however, the question asks for the two congruent sides of Triangle B. In similar triangles, the ratio of the corresponding sides must be equal. We know that the base of Triangle A is and the base of Triangle B is . We then set up a cross-multiplication using the ratio of the two bases and the ratio of $\small c$ to the side we're trying to find ( $\small x$ ), as follows:

$\small \frac{6}{12}=\frac{5}{x}$

$\small 6x=60$

$\small x=10$

Therefore, the length of the congruent sides of Triangle B is .

Compare your answer with the correct one above

Question

Isosceles triangles $\Delta ABC$ and $\Delta ABD$ share common side . $\Delta ABC$ is an obtuse triangle with sides . $\Delta ABD$ is also an obtuse isosceles triangle, where . What is the measure of $\angle AD$ ?

Answer

In order to prove triangle congruence, the triangles must have SAS, SSS, AAS, or ASA congruence. Here, we have one common side (S), and no other demonstrated congruence. Hence, we cannot guarantee that side is not one of the two congruent sides of $\Delta ABD$ , so we cannot state congruence with $\Delta ABC$ .

Compare your answer with the correct one above

Question

Triangle A and Triangle B are similar isosceles triangles. Triangle A's sides measure , , and . Two of the angles in Triangle A each measure $\small 83^o$ . Triangle B's sides measure , , and . What is the measure of the smallest angle in Triangle B?

Answer

Because the interior angles of a triangle add up to $\small 180^o$ , and two of Triangle A's interior angles measure $\small 83^o$ , we must simply add the two given angles and subtract from $\small 180^o$ to find the missing angle:

$\small 83^o+83^o=166^o$

$\small 180^o-166^o=14^o$

Therefore, the missing angle (and the smallest) from Triangle A measures $\small 14^o$ . If the two triangles are similar, their interior angles must be congruent, meaning that the smallest angle is Triangle B is also $\small 14^o$ .

The side measurements presented in the question are not needed to find the answer!

Compare your answer with the correct one above

Question

Triangle A and Triangle B are similar isosceles triangles. Triangle A has a base of and a height of . Triangle B has a base of . What is the length of Triangle B's two congruent sides?

Answer

We must first find the length of the congruent sides in Triangle A. We do this by setting up a right triangle with the base and the height, and using the Pythagorean Theorem to solve for the missing side ( $\small c$ ). Because the height line cuts the base in half, however, we must use for the length of the base's side in the equation instead of . This is illustrated in the figure below:

Using the base of $\small 3cm$ and the height of $\small 4cm$ , we use the Pythagorean Theorem to solve for $\small c$ :

$\small 3^2+4^2=c^2$

$\small 9+16=c^2$

$\small 25=c^2$

$\small c=5$

Therefore, the two congruent sides of Triangle A measure $\small 5cm$ ; however, the question asks for the two congruent sides of Triangle B. In similar triangles, the ratio of the corresponding sides must be equal. We know that the base of Triangle A is and the base of Triangle B is . We then set up a cross-multiplication using the ratio of the two bases and the ratio of $\small c$ to the side we're trying to find ( $\small x$ ), as follows:

$\small \frac{6}{12}=\frac{5}{x}$

$\small 6x=60$

$\small x=10$

Therefore, the length of the congruent sides of Triangle B is .

Compare your answer with the correct one above

Question

Isosceles triangles $\Delta ABC$ and $\Delta ABD$ share common side . $\Delta ABC$ is an obtuse triangle with sides . $\Delta ABD$ is also an obtuse isosceles triangle, where . What is the measure of $\angle AD$ ?

Answer

In order to prove triangle congruence, the triangles must have SAS, SSS, AAS, or ASA congruence. Here, we have one common side (S), and no other demonstrated congruence. Hence, we cannot guarantee that side is not one of the two congruent sides of $\Delta ABD$ , so we cannot state congruence with $\Delta ABC$ .

Compare your answer with the correct one above

Question

Triangle A and Triangle B are similar isosceles triangles. Triangle A's sides measure , , and . Two of the angles in Triangle A each measure $\small 83^o$ . Triangle B's sides measure , , and . What is the measure of the smallest angle in Triangle B?

Answer

Because the interior angles of a triangle add up to $\small 180^o$ , and two of Triangle A's interior angles measure $\small 83^o$ , we must simply add the two given angles and subtract from $\small 180^o$ to find the missing angle:

$\small 83^o+83^o=166^o$

$\small 180^o-166^o=14^o$

Therefore, the missing angle (and the smallest) from Triangle A measures $\small 14^o$ . If the two triangles are similar, their interior angles must be congruent, meaning that the smallest angle is Triangle B is also $\small 14^o$ .

The side measurements presented in the question are not needed to find the answer!

Compare your answer with the correct one above

Question

Triangle A and Triangle B are similar isosceles triangles. Triangle A has a base of and a height of . Triangle B has a base of . What is the length of Triangle B's two congruent sides?

Answer

We must first find the length of the congruent sides in Triangle A. We do this by setting up a right triangle with the base and the height, and using the Pythagorean Theorem to solve for the missing side ( $\small c$ ). Because the height line cuts the base in half, however, we must use for the length of the base's side in the equation instead of . This is illustrated in the figure below:

Using the base of $\small 3cm$ and the height of $\small 4cm$ , we use the Pythagorean Theorem to solve for $\small c$ :

$\small 3^2+4^2=c^2$

$\small 9+16=c^2$

$\small 25=c^2$

$\small c=5$

Therefore, the two congruent sides of Triangle A measure $\small 5cm$ ; however, the question asks for the two congruent sides of Triangle B. In similar triangles, the ratio of the corresponding sides must be equal. We know that the base of Triangle A is and the base of Triangle B is . We then set up a cross-multiplication using the ratio of the two bases and the ratio of $\small c$ to the side we're trying to find ( $\small x$ ), as follows:

$\small \frac{6}{12}=\frac{5}{x}$

$\small 6x=60$

$\small x=10$

Therefore, the length of the congruent sides of Triangle B is .

Compare your answer with the correct one above

Question

Isosceles triangles $\Delta ABC$ and $\Delta ABD$ share common side . $\Delta ABC$ is an obtuse triangle with sides . $\Delta ABD$ is also an obtuse isosceles triangle, where . What is the measure of $\angle AD$ ?

Answer

In order to prove triangle congruence, the triangles must have SAS, SSS, AAS, or ASA congruence. Here, we have one common side (S), and no other demonstrated congruence. Hence, we cannot guarantee that side is not one of the two congruent sides of $\Delta ABD$ , so we cannot state congruence with $\Delta ABC$ .

Compare your answer with the correct one above

Question

Triangle A and Triangle B are similar isosceles triangles. Triangle A's sides measure , , and . Two of the angles in Triangle A each measure $\small 83^o$ . Triangle B's sides measure , , and . What is the measure of the smallest angle in Triangle B?

Answer

Because the interior angles of a triangle add up to $\small 180^o$ , and two of Triangle A's interior angles measure $\small 83^o$ , we must simply add the two given angles and subtract from $\small 180^o$ to find the missing angle:

$\small 83^o+83^o=166^o$

$\small 180^o-166^o=14^o$

Therefore, the missing angle (and the smallest) from Triangle A measures $\small 14^o$ . If the two triangles are similar, their interior angles must be congruent, meaning that the smallest angle is Triangle B is also $\small 14^o$ .

The side measurements presented in the question are not needed to find the answer!

Compare your answer with the correct one above

Question

Triangle A and Triangle B are similar isosceles triangles. Triangle A has a base of and a height of . Triangle B has a base of . What is the length of Triangle B's two congruent sides?

Answer

We must first find the length of the congruent sides in Triangle A. We do this by setting up a right triangle with the base and the height, and using the Pythagorean Theorem to solve for the missing side ( $\small c$ ). Because the height line cuts the base in half, however, we must use for the length of the base's side in the equation instead of . This is illustrated in the figure below:

Using the base of $\small 3cm$ and the height of $\small 4cm$ , we use the Pythagorean Theorem to solve for $\small c$ :

$\small 3^2+4^2=c^2$

$\small 9+16=c^2$

$\small 25=c^2$

$\small c=5$

Therefore, the two congruent sides of Triangle A measure $\small 5cm$ ; however, the question asks for the two congruent sides of Triangle B. In similar triangles, the ratio of the corresponding sides must be equal. We know that the base of Triangle A is and the base of Triangle B is . We then set up a cross-multiplication using the ratio of the two bases and the ratio of $\small c$ to the side we're trying to find ( $\small x$ ), as follows:

$\small \frac{6}{12}=\frac{5}{x}$

$\small 6x=60$

$\small x=10$

Therefore, the length of the congruent sides of Triangle B is .

Compare your answer with the correct one above

Question

Isosceles triangles $\Delta ABC$ and $\Delta ABD$ share common side . $\Delta ABC$ is an obtuse triangle with sides . $\Delta ABD$ is also an obtuse isosceles triangle, where . What is the measure of $\angle AD$ ?

Answer

In order to prove triangle congruence, the triangles must have SAS, SSS, AAS, or ASA congruence. Here, we have one common side (S), and no other demonstrated congruence. Hence, we cannot guarantee that side is not one of the two congruent sides of $\Delta ABD$ , so we cannot state congruence with $\Delta ABC$ .

Compare your answer with the correct one above

Question

Triangle A and Triangle B are similar isosceles triangles. Triangle A's sides measure , , and . Two of the angles in Triangle A each measure $\small 83^o$ . Triangle B's sides measure , , and . What is the measure of the smallest angle in Triangle B?

Answer

Because the interior angles of a triangle add up to $\small 180^o$ , and two of Triangle A's interior angles measure $\small 83^o$ , we must simply add the two given angles and subtract from $\small 180^o$ to find the missing angle:

$\small 83^o+83^o=166^o$

$\small 180^o-166^o=14^o$

Therefore, the missing angle (and the smallest) from Triangle A measures $\small 14^o$ . If the two triangles are similar, their interior angles must be congruent, meaning that the smallest angle is Triangle B is also $\small 14^o$ .

The side measurements presented in the question are not needed to find the answer!

Compare your answer with the correct one above

Question

Triangle A and Triangle B are similar isosceles triangles. Triangle A has a base of and a height of . Triangle B has a base of . What is the length of Triangle B's two congruent sides?

Answer

We must first find the length of the congruent sides in Triangle A. We do this by setting up a right triangle with the base and the height, and using the Pythagorean Theorem to solve for the missing side ( $\small c$ ). Because the height line cuts the base in half, however, we must use for the length of the base's side in the equation instead of . This is illustrated in the figure below:

Using the base of $\small 3cm$ and the height of $\small 4cm$ , we use the Pythagorean Theorem to solve for $\small c$ :

$\small 3^2+4^2=c^2$

$\small 9+16=c^2$

$\small 25=c^2$

$\small c=5$

Therefore, the two congruent sides of Triangle A measure $\small 5cm$ ; however, the question asks for the two congruent sides of Triangle B. In similar triangles, the ratio of the corresponding sides must be equal. We know that the base of Triangle A is and the base of Triangle B is . We then set up a cross-multiplication using the ratio of the two bases and the ratio of $\small c$ to the side we're trying to find ( $\small x$ ), as follows:

$\small \frac{6}{12}=\frac{5}{x}$

$\small 6x=60$

$\small x=10$

Therefore, the length of the congruent sides of Triangle B is .

Compare your answer with the correct one above

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