How to find an angle in an acute / obtuse triangle - High School Math

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Question

In the triangle below, AB=BC (figure is not to scale) . If angle A is 41°, what is the measure of angle B?

A (Angle A = 41°)

B C

Answer

If angle A is 41°, then angle C must also be 41°, since AB=BC. So, the sum of these 2 angles is:

41° + 41° = 82°

Since the sum of the angles in a triangle is 180°, you can find out the measure of the remaining angle by subtracting 82 from 180:

180° - 82° = 98°

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Question

If the average of the measures of two angles in a triangle is 75o, what is the measure of the third angle in this triangle?

Answer

The sum of the angles in a triangle is 180o: a + b + c = 180

In this case, the average of a and b is 75:

(a + b)/2 = 75, then multiply both sides by 2

(a + b) = 150, then substitute into first equation

150 + c = 180

c = 30

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Question

Points A, B, C, D are collinear. The measure of ∠ DCE is 130° and of ∠ AEC is 80°. Find the measure of ∠ EAD.

Answer

To solve this question, you need to remember that the sum of the angles in a triangle is 180°. You also need to remember supplementary angles. If you know what ∠ DCE is, you also know what ∠ ECA is. Hence you know two angles of the triangle, 180°-80°-50°= 50°.

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Question

Which of the following can NOT be the angles of a triangle?

Answer

In a triangle, there can only be one obtuse angle. Additionally, all the angle measures must add up to 180.

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Question

Let the measures, in degrees, of the three angles of a triangle be x, y, and z. If y = 2z, and z = 0.5x - 30, then what is the measure, in degrees, of the largest angle in the triangle?

Answer

The measures of the three angles are x, y, and z. Because the sum of the measures of the angles in any triangle must be 180 degrees, we know that x + y + z = 180. We can use this equation, along with the other two equations given, to form this system of equations:

x + y + z = 180

y = 2z

z = 0.5x - 30

If we can solve for both y and x in terms of z, then we can substitute these values into the first equation and create an equation with only one variable.

Because we are told already that y = 2z, we alreay have the value of y in terms of z.

We must solve the equation z = 0.5x - 30 for x in terms of z.

Add thirty to both sides.

z + 30 = 0.5x

Mutliply both sides by 2

2(z + 30) = 2z + 60 = x

x = 2z + 60

Now we have the values of x and y in terms of z. Let's substitute these values for x and y into the equation x + y + z = 180.

(2z + 60) + 2z + z = 180

5z + 60 = 180

5z = 120

z = 24

Because y = 2z, we know that y = 2(24) = 48. We also determined earlier that x = 2z + 60, so x = 2(24) + 60 = 108.

Thus, the measures of the three angles of the triangle are 24, 48, and 108. The question asks for the largest of these measures, which is 108.

The answer is 108.

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Question

Angles x, y, and z make up the interior angles of a scalene triangle. Angle x is three times the size of y and 1/2 the size of z. How big is angle y.

Answer

The answer is 18

We know that the sum of all the angles is 180. Using the rest of the information given we can write the other two equations:

x + y + z = 180

x = 3y

2x = z

We can solve for y and z in the second and third equations and then plug into the first to solve.

x + (1/3)x + 2x = 180

3\[x + (1/3)x + 2x = 180\]

3x + x + 6x = 540

10x = 540

x = 54

y = 18

z = 108

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Question

If the average (arithmetic mean) of two noncongruent angles of an isosceles triangle is , which of the following is the measure of one of the angles of the triangle?

Answer

Since the triangle is isosceles, we know that 2 of the angles (that sum up to 180) must be equal. The question states that the noncongruent angles average 55°, thus providing us with a system of two equations:

$\frac{x+y}{2}=55^o$

Solving for x and y by substitution, we get x = 70° and y = 40° (which average out to 55°).

70 + 70 + 40 equals 180 also checks out.

Since 70° is not an answer choice for us, we know that the 40° must be one of the angles.

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Question

Points A, B, and C are collinear (they lie along the same line). The measure of angle CAD is $30^{\circ}$ . The measure of angle CBD is $60^{\circ}$ . The length of segment $\overline{AD}$ is 4.

Find the measure of $\dpi{100} \small \angle ADB$ .

Answer

The measure of $\dpi{100} \small \angle ADB$ is $30^{\circ}$ . Since $\dpi{100} \small A$ , $\dpi{100} \small B$ , and $\dpi{100} \small C$ are collinear, and the measure of $\dpi{100} \small \angle CBD$ is $60^{\circ}$ , we know that the measure of $\dpi{100} \small \angle ABD$ is $120^{\circ}$ .

Because the measures of the three angles in a triangle must add up to $180^{\circ}$ , and two of the angles in triangle $\dpi{100} \small ABD$ are $30^{\circ}$ and $120^{\circ}$ , the third angle, $\dpi{100} \small \angle ADB$ , is $30^{\circ}$ .

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Question

Triangle ABC has angle measures as follows:

$\dpi{100} \small m\angle ABC=4x+3$

$\dpi{100} \small m\angle ACB=2x+6$

$\dpi{100} \small m\angle BAC=3x$

What is $\dpi{100} \small m\angle BAC$ ?

Answer

The sum of the measures of the angles of a triangle is 180.

Thus we set up the equation $\dpi{100} \small 4x+3+2x+6+3x=180$

After combining like terms and cancelling, we have $\dpi{100} \small 9x=171\rightarrow x=19$

Thus $\dpi{100} \small m\angle BAC=3x=57$

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Question

The base angle of an isosceles triangle is $27^{\circ}$ . What is the vertex angle?

Answer

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

Solve the equation $27+27+x=180$ for x to find the measure of the vertex angle.

x = 180 - 27 - 27

x = 126

Therefore the measure of the vertex angle is $126^{\circ}$ .

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Question

The base angle of an isosceles triangle is five more than twice 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 $x$ = the vertex angle and $2x+5$ = the base angle

So the equation to solve becomes $x+(2x+5)+(2x+5)=180$

Thus the vertex angle is 34 and the base angles are 73.

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Question

Two interior angles in an obtuse triangle measure $123^{\circ}$ and $11^{\circ}$ . What is the measurement of the third angle.

Answer

Interior angles of a triangle always add up to 180 degrees.

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Question

The base angle of an isosceles triangle is 15 less than three times the vertex angle. What is the vertex angle?

Answer

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

Let = vertex angle and = base angle

So the equation to solve becomes .

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Question

The base angle of an isosceles triangle is ten less than twice the vertex angle. What is the vertex 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

So the equation to solve becomes

So the vertex angle is 40 and the base angles is 70

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Question

The base angle of an isosceles triangle is 10 more than twice the vertex angle. What is the vertex angle?

Answer

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

Let = the vertex angle and = the base angle

So the equation to solve becomes

The vertex angle is 32 degrees and the base angle is 74 degrees

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Question

In an isosceles triangle, the vertex angle is 15 less than the base 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 = base angle and = vertex angle

So the equation to solve becomes

Thus, 65 is the base angle and 50 is the vertex angle.

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Question

In a given triangle, the angles are in a ratio of 1:3:5. What size is the middle angle?

Answer

Since the sum of the angles of a triangle is $180^{\circ}$ , and given that the angles are in a ratio of 1:3:5, let the measure of the smallest angle be , then the following expression could be written:

$x+3x+5x=180$

$9x=180$

$x=20$

If the smallest angle is 20 degrees, then given that the middle angle is in ratio of 1:3, the middle angle would be 3 times as large, or 60 degrees.

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Question

In an isosceles triangle the vertex angle is half the base angle. What is the vertex angle?

Answer

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

Let $x$ = base angle and $0.5x$ = vertex angle

So the equation to solve becomes $x+x+0.5x=180$ , thus $x=72$ is the base angle and $0.5x=36$ is the vertex angle.

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Question

In the picture above, $\overline{AB}$ is a straight line segment. Find the value of .

Answer

A straight line segment has 180 degrees. Therefore, the angle that is not labelled must have:

$180-110 = 70^{\circ}$

We know that the sum of the angles in a triangle is 180 degrees. As a result, we can set up the following algebraic equation:

Subtract 70 from both sides:

Divide by 2:

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Question

Solve for . (Not drawn to scale).

Answer

The angles of a triangle must add to 180o. In the triangle to the right, we know one angle and can find another using supplementary angles.

Now we only need to solve for .

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