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High School Math : Plane Geometry

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Example Questions

Example Question #1 : Acute / Obtuse Triangles

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

Possible Answers:

$45^{\circ}$

$60^{\circ}$

$75^{\circ}$

$20^{\circ}$

$90^{\circ}$

Correct answer:

$60^{\circ}$

Explanation:

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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Example Question #4 : Isosceles Triangles

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

Possible Answers:

Correct answer:

Explanation:

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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Example Question #5 : Isosceles Triangles

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

Possible Answers:

$62$

$34$

$47$

$55$

$73$

Correct answer:

$73$

Explanation:

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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Example Question #6 : Isosceles Triangles

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

Possible Answers:

Correct answer:

Explanation:

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

Let = vertex angle and 3x-15 = base angle

So the equation to solve becomes x+(3x-15)+(3x-15)=180 .

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Example Question #4 : Isosceles Triangles

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

Possible Answers:

$70^{\circ}$

$35^{\circ}$

$20^{\circ}$

$40^{\circ}$

$65^{\circ}$

Correct answer:

$40^{\circ}$

Explanation:

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

Let = vertex angle and 2x - 10 = base angle

So the equation to solve becomes x + (2x - 10) + (2x - 10) = 180

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

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Example Question #5 : How To Find An Angle In An Acute / Obtuse Triangle

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

Possible Answers:

$60^{\circ}$

$50^{\circ}$

$32^{\circ}$

$74^{\circ}$

$45^{\circ}$

Correct answer:

$32^{\circ}$

Explanation:

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

Let = the vertex angle and 2x + 10 = the base angle

So the equation to solve becomes x + (2x +10) + (2x +10) = 180

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

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Example Question #6 : How To Find An Angle In An Acute / Obtuse Triangle

In an isosceles triangle, the vertex angle is 15 less than the base angle. What is the base angle?

Possible Answers:

Correct answer:

Explanation:

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

Let = base angle and x - 15 = vertex angle

So the equation to solve becomes (x - 15) + x + x = 180

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

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Example Question #11 : Isosceles Triangles

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

Possible Answers:

$108$

$54$

$45$

$36$

$72$

Correct answer:

$36$

Explanation:

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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Example Question #1 : Acute / Obtuse Triangles

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

Possible Answers:

75°

50°

40°

30°

65°

Correct answer:

30°

Explanation:

The sum of the angles in a triangle is 180^o: 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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Example Question #2 : Acute / Obtuse Triangles

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

Possible Answers:

30.5, 40.1, 109.4

45, 90, 100

45, 45, 90

30, 60, 90

1, 2, 177

Correct answer:

45, 90, 100

Explanation:

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

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High School Math : Plane Geometry

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #1 : Acute / Obtuse Triangles

Example Question #4 : Isosceles Triangles

Example Question #5 : Isosceles Triangles

Example Question #6 : Isosceles Triangles

Example Question #4 : Isosceles Triangles

Example Question #5 : How To Find An Angle In An Acute / Obtuse Triangle

Example Question #6 : How To Find An Angle In An Acute / Obtuse Triangle

Example Question #11 : Isosceles Triangles

Example Question #1 : Acute / Obtuse Triangles

Example Question #2 : Acute / Obtuse Triangles

All High School Math Resources

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