Chapter 1: Triangles and Circles
Exercises: 1.1 Triangles and Angles
SKILLS
Practice each skill in the Homework Problems listed.
- Sketch a triangle with given properties #1–6
- Find an unknown angle in a triangle #7–12, 17–20
- Find angles formed by parallel lines and a transversal #13–16, 35–44
- Find exterior angles of a triangle #21–24
- Find angles in isosceles, equilateral, and right triangles #25–34
- State reasons for conclusions #45–48
Suggested Problems
Problems: #4, 8, 12, 14, 18, 22, 28, 34, 38, 46.
Exercises for 1.1 Triangles and Angles
EXERCISE GROUP
2. A scalene triangle with one obtuse angle (Scalene means three unequal sides.)
3. A right triangle with legs [latex]4[/latex] and [latex]7[/latex]
4. An isosceles right triangle
5. An isosceles triangle with one obtuse angle
6. A right triangle with one angle [latex]20°[/latex]
EXERCISE GROUP
7.
8.
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12.
13.
14.
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16.
17.
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20.
EXERCISE GROUP
21.
22.
23.
In parts (a) and (b), find the exterior angle [latex]\phi[/latex].
- Find an algebraic expression for [latex]\phi[/latex]
- Use your answer to part (c) to write a rule for finding an exterior angle of a triangle.
24.
- Find the three exterior angles of the triangle. What is the sum of the exterior angles?
- Write an algebraic expression for each exterior angle in terms of one of the angles of the triangle. What is the sum of the exterior angles?
EXERCISE GROUP
In Problems 25 and 26, the figures inscribed are regular polygons, which means that all their sides are the same length, and all the angles have the same measure. Find the angles [latex]\theta[/latex] and [latex]\phi[/latex].
25.
26.
EXERCISE GROUP
In problems 27 and 28, triangle ABC is equilateral. Find the unknown angles.
27.
28.
29.
a. [latex]2\theta + 2\phi =[/latex]
b. [latex]\theta + \phi =[/latex]
c. [latex]\triangle ABC[/latex] is
30.
Find [latex]\alpha[/latex] and [latex]\beta[/latex]
31.
- Explain why [latex]\angle OAB[/latex] and [latex]\angle ABO[/latex] are equal in measure.
- Explain why [latex]\angle OBC[/latex] and [latex]\angle BCO[/latex] are equal in measure.
- Explain why [latex]\angle ABC[/latex] is a right angle. (Hint: Use Problem 29.)
32.
- Compare [latex]\theta[/latex] with [latex]\alpha + \beta[/latex] (Hint: What do you know about supplementary angles and the sum of angles in a triangle?)
- Compare [latex]\alpha[/latex] and [latex]\beta[/latex]
- Explain why the inscribed angle [latex]\angle BAO[/latex] is half the size of the central angle [latex]\angle BOD[/latex]
EXERCISE GROUP
33.
Find [latex]\alpha[/latex] and [latex]\beta[/latex]
34.
Find [latex]\alpha[/latex] and [latex]\beta[/latex]
EXERCISE GROUP
35.
36.
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38.
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44.
45.
- Among the angles labeled 1 through 5 in the figure at right, find two pairs of equal angles.
- [latex]\angle 4 + \angle 2 + \angle 5 =[/latex]
- Use parts (a) and (b) to explain why the sum of the angles of a triangle is [latex]180^{\circ}[/latex]
46.
- In the figure below, find [latex]\theta[/latex] and justify your answer.
- Write an algebraic expression for [latex]\theta[/latex] in the figure below.
47.
ABCD is a rectangle. The diagonals of a rectangle bisect each other. In the figure, [latex]\angle AQD = 130^{\circ}[/latex]. Find the angles labeled 1 through 5 in order, and give a reason for each answer.
48.
A tangent meets the radius of a circle at a right angle. In the figure, [latex]\angle AOB = 140^{\circ}[/latex]. Find the angles labeled 1 through 5 in order, and give a reason for each answer.
