Acute / Obtuse Isosceles Triangles - ACT Math

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Question

Points A and B lie on a circle centered at Z, where central angle <AZB measures 140°. What is the measure of angle <ZAB?

Answer

Because line segments ZA and ZB are radii of the circle, they must have the same length. That makes triangle ABZ an isosceles triangle, with <ZAB and <ZBA having the same measure. Because the three angles of a triangle must sum to 180°, you can express this in the equation:

140 + 2x = 180 --> 2x = 40 --> x = 20

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Question

In triangle ABC, Angle A = x degrees, Angle B = 2x degrees, and Angle C = 3x+30 degrees. How many degrees is Angle B?

Answer

Because the interior angles of a triangle add up to 180°, we can create an equation using the variables given in the problem: x+2x+(3x+30)=180. This simplifies to 6X+30=180. When we subtract 30 from both sides, we get 6x=150. Then, when we divide both sides by 6, we get x=25. Because Angle B=2x degrees, we multiply 25 times 2. Thus, Angle B is equal to 50°. If you got an answer of 25, you may have forgotten to multiply by 2. If you got 105, you may have found Angle C instead of Angle B.

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Question

Triangle FGH has equal lengths for FG and GH; what is the measure of ∠F, if ∠G measures 40 degrees?

Answer

It's good to draw a diagram for this; we know that it's an isosceles triangle; remember that the angles of a triangle total 180 degrees.

Angle G for this triangle is the one angle that doesn't correspond to an equal side of the isosceles triangle (opposite side to the angle), so that means ∠F = ∠H, and that ∠F + ∠H + 40 = 180,

By substitution we find that ∠F * 2 = 140 and angle F = 70 degrees.

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Question

The vertex angle of an isosceles triangle is $54^{\circ}$ . What is the base angle?

Answer

An isosceles triangle has two congruent base angles and one vertex angle. Each triangle contains $180^{\circ}$ . Let = base angle, so the equation becomes . Solving for gives

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Question

In an isosceles triangle the base angle is five less than twice the vertex angle. What is the sum of the vertex angle and the base angle?

Answer

Every triangle has 180 degrees. An isosceles triangle has one vertex angle and two congruent base angles.

Let = the vertex angle

and = base angle

So the equation to solve becomes

or

Thus the vertex angle is 38 and the base angle is 71 and their sum is 109.

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Question

An isosceles triangle has a base angle that is six more than three times the vertex angle. What is the base angle?

Answer

Every triangle has 180 degrees. An isosceles triangle has one vertex angle and two congruent base angles.

Let = vertex angle and = base angle.

Then the equation to solve becomes

or

.

Solving for gives a vertex angle of 24 degrees and a base angle of 78 degrees.

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Question

The base angle of an isosceles triangle is thirteen more than three times the vertex angle. What is the difference between the vertex angle and the base angle?

Answer

Every triangle has . An isosceles triangle has one vertex ange, and two congruent base angles.

Let be the vertex angle and be the base angle.

The equation to solve becomes , since the base angle occurs twice.

Now we can solve for the vertex angle.

The difference between the vertex angle and the base angle is .

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Question

Sides and in this triangle are equal. What is the measure of $\angle A$ ?

Answer

This triangle has an angle of $65^{\circ}$ . We also know it has another angle of $65^{\circ}$ at $\angle ABC$ because the two sides are equal. Adding those two angles together gives us $130^{\circ}$ total. Since a triangle has $180^{\circ}$ total, we subtract 130 from 180 and get 50.

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Question

A particular acute isosceles triangle has an internal angle measuring $40^{\circ}$ . Which of the following must be the other two angles?

Answer

By definition, an acute isosceles triangle will have at least two sides (and at least two corresponding angles) that are congruent, and no angle will be greater than $90^{\circ}$ . Addtionally, like all triangles, the three angles will sum to $180^{\circ}$ . Thus, of our two answers which sum to $180^{\circ}$ , only $70^{\circ}, 70^{\circ}$ is valid, as $40^{\circ}, 100^{\circ}$ would violate the "acute" part of the definition.

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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

There are two obtuse triangles. The obtuse angle of triangle one is $101^{\circ}$ . The sum of two angles in the second triangle is $80^{\circ}$ . When are these two triangles congruent?

Answer

In order for two obtuse triangles to be congruent, the sum of the two smaller angles must equal the sum of the two smaller angles of the second triangle. That is, excluding the obtuse angle.

The first triangle has an obtuse angle of $101^{\circ}$ . That means the sum of the other two angles is $79^{\circ}$ . The sum of the corresponding angles in triangle 2 is $80^{\circ}$ . Therefore, because $79^{\circ}$ is not equal to $80^{\circ}$ , the two obtuse triangles cannot be congruent.

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Question

What is the area of an isosceles triangle with a vertex of degrees and two sides equal to ?

Answer

Based on the description of your triangle, you can draw the following figure:

You can do this because you know:

The two equivalent sides are given.

Since a triangle is degrees, you have only or degrees left for the two angles of equal size. Therefore, those two angles must be degrees and degrees.

Now, based on the properties of an isosceles triangle, you can draw the following as well:

Based on your standard reference triangle, you know:

$\frac{h}{4}=\frac{1}{2}$

Therefore, is .

This means that is $2\sqrt{3}$ and the total base of the triangle is $4\sqrt{3}$ .

Now, the area of the triangle is:

$\frac{1}{2}bh$ or $\frac{1}{2}24\sqrt{3}=4\sqrt{3}:cm^2$

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Question

An isosceles triangle has a height of and a base of . What is its area?

Answer

Use the formula for area of a triangle:

$A=\frac{1}{2}(base)(height)$

$A=\frac{1}{2}(4)(7)=14$

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Question

An isosceles triangle has a base length of and a height that is twice its base length. What is the area of this triangle?

Answer

1. Find the height of the triangle:

$height=2\cdot base$

2. Use the formula for area of a triangle:

$A=\frac{1}{2}(b)(h)$

$A=\frac{1}{2}(10)(20)=100$

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Question

An isosceles triangle has a base of $12\ cm$ and an area of $42\ cm^{2}$ . What must be the height of this triangle?

Answer

$A=\frac{1}{2}bh$ .

$6x=42$

$x=7$

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Question

What is the height of an isosceles triangle which has a base of and an area of ?

Answer

The area of a triangle is given by the equation:

$\small A=\frac{1}{2}bh$

In this case, we are given the area ( $\small A$ ) and the base ( $\small b$ ) and are asked to solve for height ( $\small h$ ).

To do this, we must plug in the given values for $\small A$ and $\small b$ , which gives the following:

$\small 24;cm^2=\frac{1}{2}(4; cm)h$

We then must multiply the right side, and then divide the entire equation by 2, in order to solve for $\small h$ :

$\small 24;cm^2=2;cm(h)$

$\small \frac{24; cm^2}{2; cm}=\frac{2; cm(h)}{2; cm}$

$\small 12;cm=h$

$\small h=12;cm$

Therefore, the height of the triangle is $\small 12;cm$ .

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Question

A triangle has a perimeter of inches with one side of length inches. If the remaining two sides have lengths in a ratio of , what is length of the shortest side of the triangle?

Answer

The answer is .

Since we know that the permieter is inches and one side is inches, it can be determined that the remaining two sides must combine to be inches. The ratio of the remaining two sides is which means 3 parts : 4 parts or 7 parts combined. We can then set up the equation , and divide both sides by which means . The ratio of the remaining side lengths then becomes or . We now know the 3 side lengths are $12,12,and\ 16$ .

is the shortest side and thus the answer.

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Question

In the standard coordinate plane, the points and form two vertices of an isosceles triangle. Which of the following points could be the third vertex?

Answer

To form an isosceles triangle here, we need to create a third vertex whose coordinate is between and . If a vertex is placed at , the distance from to this point will be $\sqrt{17}$ . The distance from to this point will be the same.

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