How to find an angle in an acute / obtuse triangle

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ACT Math › How to find an angle in an acute / obtuse triangle

Questions 1 - 5
1

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

60^{\circ}

20^{\circ}

90^{\circ}

45^{\circ}

75^{\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.

2

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°)

Act_math_108_02

B C

41

82

90

98

Explanation

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°

3

Triangles

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.

30^{\circ}

60^{\circ}

90^{\circ}

15^{\circ}

45^{\circ}

Explanation

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}.

4

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

46^{\circ}

50^{\circ}

57^{\circ}

123^{\circ}

104^{\circ}

Explanation

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

5

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

Screen_shot_2013-03-18_at_3.27.08_pm

50°

60°

70°

80°

Explanation

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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