Intermediate Geometry : How to find an angle in an acute / obtuse triangle

Study concepts, example questions & explanations for Intermediate Geometry

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Example Questions

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Example Question #21 : How To Find An Angle In An Acute / Obtuse Triangle

Two of the exterior angles of a triangle, taken at different vertices, measure  and . Is the triangle acute, right, or obtuse?

Possible Answers:

Right

Obtuse

Acute 

Correct answer:

Right

Explanation:

At a given vertex, an exterior angle and an interior angle of a triangle form a linear pair, making them supplementary - that is, their measures total . The measures of two interior angles can be calculated by subtracting the exterior angle measures from :

The triangle has two interior angles of measures  and . The sum of these measures is , thereby making them complementary. A triangle with two complementary acute angles is a right triangle.

Example Question #551 : Intermediate Geometry

True or false: It is possible for a triangle to have angles of measure , and .

Possible Answers:

True

False

Correct answer:

True

Explanation:

The sum of the measures of the angles of a triangle is . The sum of the three given angle measures is 

.

This makes the triangle possible.

Example Question #51 : Acute / Obtuse Triangles

True or false: It is possible for a triangle to have three interior angles, each of whose measures are .

Possible Answers:

True

False

Correct answer:

False

Explanation:

A triangle with three congruent angles is an equiangular - and equilateral - triangle; such an angle must have three angles that measure 

Example Question #22 : How To Find An Angle In An Acute / Obtuse Triangle

Given:  with perimeter 40;

True or false: 

Possible Answers:

False

True

Correct answer:

True

Explanation:

The perimeter of  is the sum of the lengths of its sides - that is,

The perimeter is 40, so set , and solve for :

Subtract 26 from both sides:

, so by the Isosceles Triangle Theorem, their opposite angles are congruent - that is,

.

Example Question #51 : Acute / Obtuse Triangles

 is an equilateral triangle;  is the midpoint of ; the segment  is constructed. 

True or false: .

Possible Answers:

True

False

Correct answer:

False

Explanation:

The referenced triangle is below:

Equilateral 2

In an equilateral triangle, the median from  - the segment from  to , the midpoint of the opposite side  - is also the bisector of the angle , so 

Each interior angle of an equilateral triangle, including , measures , so substitute and evaluate:

.

Example Question #551 : Intermediate Geometry

 is an equilateral triangle. Locate a point  along  and construct 

Evaluate .

Possible Answers:

Correct answer:

Explanation:

The referenced figure is below. Note that , as is the case with all of the interior angles of an equilateral triangle.

Equilateral

The interior angles of an equilateral triangle each measure . An exterior angle of a triangle has as its degree measure the sum of its remote interior angles; specifically, 

Substitute the known angle measures, and solve:

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