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Chapter 3: Problem 17

Two exterior angles of a triangle sum to \(200^{\circ} .\) What is the measure of the third exterior angle?

Short Answer

Expert verified
The measure of the third exterior angle is 160掳.

Step by step solution

01

Understand the Problem

We are given two exterior angles of a triangle which sum to 200掳. Our goal is to find the measure of the third exterior angle.
02

Recall the Property of Exterior Angles

The sum of all exterior angles of any triangle is always 360掳.
03

Set Up the Equation

Let the measure of the third exterior angle be denoted as x. According to the property of exterior angles, we can set up the equation: 200掳 + x = 360掳.
04

Solve for the Third Exterior Angle

Subtract 200掳 from both sides of the equation to isolate x: x = 360掳 - 200掳 x = 160掳.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

sum of exterior angles
In any triangle, the sum of the exterior angles is a constant value of 360掳. This property is crucial because it holds true for all triangles, regardless of their shape or size. This unique property helps us in solving various problems related to angle measurements in triangles.
For instance, in the given exercise, knowing that the sum of the exterior angles is 360掳 allowed us to set up an equation and find the measure of the unknown exterior angle.
When faced with a problem where some angles are given and others need to be calculated, remember to use this property. Sum up the known exterior angles and subtract from 360掳 to find the missing value.
triangle properties
Triangles have several important properties that help us solve problems involving their angles and sides. Here are a few key triangle properties:
  • Interior Angles: The sum of the interior angles of a triangle is always 180掳.

  • Exterior Angles: Each exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles.

  • Types of Triangles: Triangles can be classified based on their side lengths (equilateral, isosceles, scalene) or their angle measurements (acute, obtuse, right).

In our exercise, we focused on the exterior angles, a fundamental property that states the sum of all exterior angles in a triangle is always 360掳. Using this property can simplify many problems and provide quick solutions.
angle measurement
Measuring angles accurately is fundamental in geometry. Here's a quick guide to understanding angle measurements in triangles:
  • Units of Measurement: Angles are most commonly measured in degrees (掳).

  • Protractors: These tools are used to measure the degree of an angle precisely.

  • Interior vs. Exterior Angles: Interior angles are inside the triangle, adding up to 180掳. Exterior angles are outside and adjacent to an interior angle, summing to 360掳 for all three angles.

  • Calculations: When given certain angles and needing to find others, always use known properties, such as the sum of interior angles being 180掳 and exterior angles being 360掳.

In our example, we used the property of the sum of exterior angles to find that the third angle needed to complete the 360掳 was 160掳. This approach simplifies solving the problem and provides a step-by-step method to follow.

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Most popular questions from this chapter

The definition of parallel lines in this section indicated the lines must be in the same plane. This is a pivotal assumption for Euclidean geometry, the geometry in this textbook. There are other nonEuclidean geometries. These geometries are characterized by at least one contradiction of a Euclidean geometry postulate. How many non-Euclidean geometries can you find? Do a little research on how they are different from Euclidean geometry.

Solve the equation \(S=(n-2) 180^{\circ}\) for \(n\) when \(S\) is a given value. Find the number of sides of each polygon (if possible) if the given value corresponds to the number of degrees in the sum of the interior angles of a polygon. Remember that \(n\) must be a whole number greater than \(2,\) or no such polygon can exist. $$2700^{\circ}$$

Give an indirect proof of each "theorem" in Exercises 3 and 4. Premise 1: If it's not a parallelogram, then it's not a rectangle. Premise 2: If it's not a four-sided figure, then it's not a quadrilateral. Premise 3: If it's not a rectangle, then it's not a square. Premise 4: If it's not a quadrilateral, then it's not a parallelogram. Theorem: If it's a square, then it's a four-sided figure.

As the number of sides of a regular polygon increases, does each exterior angle increase or decrease?

In Exercises \(17-22\), assume that \(\ell, m,\) and \(n\) are three distinct lines in a plane and \(P\) is a point in the plane. \(If \quad\ell \perp m \quad and \quad\ell \perp n, is \quad m \| n ?\)
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