Geometry M Angle – Angles And Angle Pairs

Explanation:

If 

*

 and 

*

 , then 

*

, since two lines parallel to the same line are parallel to each other.

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If

*

, then

*

, since two same-side interior angles formed by transversal 

*

 are supplementary.

If 

*

, then 

*

, since two alternate interior angles formed by transversal 

*

 are congruent.

However, 

*

 regardless of whether 

*

 and 

*

 are parallel; they are vertical angles, and by the Vertical Angles Theorem, they must be congruent.

An isosceles triangle has an interior angle that measures 

*

. What are the measures of its other two angles?

 

*

*

*

Correct answer:

 

Explanation:

By the Isosceles Triangle Theorem, two interior angles must be congruent. However, since a triangle cannot have two obtuse interior angles, the two missing angles must be the ones that are congruent. Since the total angle measure of a triangle is 

*

 , each of the missing angles measures

*

.

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Explanation:

Obtuse angles are greater than

*

.

Scalene is a designation for triangles that have one angle greater than

*

, but this figure is not a triangle.

Acute angles are less than

*

, right angles are

*

, and straight angles are

*

.

Therefore this angle is obtuse.

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Explanation:

When two parallel lines are crossed by a third line (called the transversal), the measure of the angles follows a specific pattern. The pairs of angles inside the two lines and on opposite sides are called alternate interior angles. Alternate interior angles, such as

*

and

*

, have the same degree measure. Therefore, the measure of

*

is

*

.

Mark is training for cross country and comes across a new hill to run on. After Mark runs

*

 meters, he”s at a height of

*

 meters. What is the hill”s angle of depression when he”s at an altitude of

*

 meters?

 

Explanation:

*

Upon reading the question, we”re left with this spatial image of Mark in our heads. After adding in the given information, the image becomes more like

*

The hill Mark is running on can be seen in terms of a right triangle. This problem quickly becomes one that is asking for a mystery angle given that the two legs of the triangle are given. In order to solve for the angle of depression, we have to call upon the principles of the tangent function. Tan, Sin, or Cos are normally used when there is an angle present and the goal is to calculate one of the sides of the triangle. In this case, the circumstances are reversed. 

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Remember back to “SOH CAH TOA.” In this problem, no information is given about the hypotenuse and nor are we trying to calculate the hypotenuse. Therefore, we are left with “TOA.” If we were to check, this would work out because the angle at Mark”s feet has the information for the opposite side and adjacent side. 

Because there”s no angle given, we must use the principles behind the tan function while using a fraction composed of the given sides. This problem will be solved using arctan (sometimes denoted as 

*

). 

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