1.13: Angle Properties and Theorems (2024)

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    Find angles and line segments, and determine if shapes are congruent and lines are parallel. Understand complementary angles as angles whose sum is 90 degrees and supplementary angles as angles whose sum is 180 degrees.

    Measures of Angle Pairs

    The foul lines of a baseball diamond intersect at home plate to form a right angle. A baseball is hit from home plate and forms an angle of \(36^{\circ}\) with the third base foul line. What is the measure of the angle between the first base foul line and the bath of the baseball?

    How can you use your knowledge of angles to figure out the measure of the angle?

    In this concept, you will learn measure of angle pairs.

    Measuring Angle Pairs

    There are different types of angle pairs. Vertical angles are an angle pair formed by intersecting lines such that they are never adjacent. They have a common vertex and never share a common side. Vertical angles are equal in measure. The following diagram shows vertical angle pairs.

    \(\angle 1\) and \(\angle 2\) are vertical angles. \(m\angle 1=m\angle 2\)

    \(\angle 3\) and \(\angle 4\) are vertical angles. \(m\angle 3=m\angle 4\)

    Adjacent angles are an angle pair also formed by two intersecting lines. Adjacent angles are side by side, have a common vertex and share a common side. The following diagram shows pairs of adjacent angles.

    Each pair of adjacent angles forms a straight angle. Therefore the sum of any two adjacent angles equals \(180^{\circ}\).

    \(m\angle 1+m \angle 3= 180^{\circ}\)

    \(m\angle 2+m \angle 4= 180^{\circ}\)

    \(m\angle 2+m \angle 3= 180^{\circ}\)

    \(m\angle 1+m \angle 4= 180^{\circ}\)

    If the sum two angles is \(180^{\circ}\) then the angles are called supplementary angles. The following diagram shows two supplementary angles.

    In both diagrams, \(m\angle 1+m \angle 2= 180^{\circ}\).

    If the sum of two angles equals 90° then the angles are called complementary angles. The following diagram shows two complementary angles.

    \(m\angle 1+m \angle 2= 90^{\circ}\)

    Let’s apply all this information about angles and their measure to determine the measure of \(\angle a\), \(\angle b\), \(\angle c\) in the following diagram.

    There are four angles formed by intersecting lines. The measure of one of the angles is \(70^{\circ}\).

    First, state the relationship between the angle of \(70^{\circ}\) and \(\angle b\).

    The angle of \(70^{\circ}\). is adjacent to \(\angle b\) and the two angles form a straight angle.

    Next, express the relationship using symbols.

    \(\angle b+70^{\circ}=180^{\circ}\)

    Next, subtract 70° from both sides of the equation.

    \(\angle b+70^{\circ}=180^{\circ}\)

    \(\angle b+70^{\circ}- 70^{\circ}=180^{\circ}-70^{\circ}\)

    Then, simplify both sides of the equation.

    \(\angle b+70^{\circ}- 70^{\circ}=180^{\circ}-70^{\circ}\)

    \(\angle b = 110^{\circ}\)

    The answer is \(110^{\circ}\).

    \(m \angle b = 110^{\circ}\)

    First, state the relationship between the angle of \(70^{\circ}\) and \(\angle a\).

    The angle of \(70^{\circ}\) and \(\angle a\) are vertical angles and are equal in measure.

    Next, express the relationship using symbols.

    \(m\angle a=70^{\circ}\)

    The answer is \(70^{\circ}\).

    \(m\angle a=70^{\circ}\)

    First state the relationship between the angle of \(70^{\circ}\) and \(\angle c\).

    The angle of \(70^{\circ}\) is adjacent to \(\angle c\) and the two angles form a straight angle.

    Next, express the relationship using symbols.

    \(\angle c+70^{\circ}=180^{\circ}\)

    Next, subtract \(70^{\circ}\) from both sides of the equation.

    \(\angle c+70^{\circ}=180^{\circ}\)

    \(\angle c+70^{\circ}-70^{\circ}=180^{\circ} -70^{\circ}\)

    Then, simplify both sides of the equation.

    \(\angle c+70^{\circ}-70^{\circ}=180^{\circ} -70^{\circ}\)

    \(\angle c=110^{\circ}\)

    The answer is \(110^{\circ}\).

    \(m \angle c=110^{\circ}\)

    Example \(\PageIndex{1}\)

    Earlier, you were given a problem about the baseball field and the foul lines.

    The angle between the path of the ball and the first base foul line needs to be figured out. This can be done using complementary angles.

    Solution

    First, draw a diagram to model the problem.

    Next, state the relationship between \(36^{\circ}\) and \(\angle x\).

    \(36^{\circ}\) and \(\angle x\) are complementary angles. The sum of the angles is \(90^{\circ}\).

    Next, express the relationship using symbols.

    \(36^{\circ}+\angle x=90^{\circ}\)

    Next, subtract 36° from both sides of the equation.

    \(36^{\circ}+\angle x=90^{\circ}\)

    \(36^{\circ}-36^{\circ}+\angle x=90^{\circ}-36^{\circ}\)

    Then, simplify both sides of the equation.

    \(36^{\circ}-36^{\circ}+\angle x=90^{\circ}-36^{\circ}\)

    \(\angle x = 54^{\circ}\)

    The answer is \(54^{\circ}\).

    An angle of \(54^{\circ}\) is made between the first base foul line and the path of the baseball.

    Example \(\PageIndex{2}\)

    If the following angles are complementary, find the measure of the missing angle.

    \(\angle A=37^{\circ}\) then \(\angle B=\)?

    Solution

    First, draw a diagram to model the problem.

    Next, state the relationship between \(\angle A\) and \(\angle B\).

    \(\angle A\) and \(\angle B\) are complementary angles. The sum of the angles is \(90^{\circ}\).

    Next, express the relationship using symbols.

    \(\angle A+ \angle B=90^{\circ}\)

    Next, substitute the measure of \(\angle A\) into the equation.

    \(37^{\circ}+ \angle B=90^{\circ}\)

    Next, subtract \(37^{\circ}\) from both sides of the equation.

    \(37^{\circ}+ \angle B=90^{\circ}\)

    \(37^{\circ}- 37^{\circ}+ \angle B=90^{\circ}- 37^{\circ}\)

    Then, simplify both sides of the equation.

    \(37^{\circ}- 37^{\circ}+ \angle B=90^{\circ}- 37^{\circ}\)

    \(\angle B =53^{\circ}\)

    The answer is \(53^{\circ}\).

    \(m \angle B =53^{\circ}\)

    Example \(\PageIndex{3}\)

    If the following angles are supplementary, find the measure of the missing angle.

    \(\angle A=102^{\circ}\) then \(\angle B=\)?

    Solution

    First, draw a diagram to model the problem.

    Next, state the relationship between \(\angle A\) and \(\angle B\).

    \(\angle A\) and \(\angle B\) are supplementary angles. The sum of the angles is 180°.

    Next, express the relationship using symbols.

    \(\angle A+ \angle B=180^{\circ}\)

    Next, substitute the measure of \(\angle A\) into the equation.

    \(102^{\circ}+\angle B=180^{\circ}\)

    Next, subtract \(102^{\circ}\) from both sides of the equation.

    \(102^{\circ}+\angle B=180^{\circ}\)

    \(102^{\circ}-102^{\circ}+\angle B=180^{\circ}-102^{\circ}\)

    Then, simplify both sides of the equation.

    \(102^{\circ}-102^{\circ}+\angle B=180^{\circ}-102^{\circ}\)

    \(\angle B=78^{\circ}\)

    The answer is \(78^{\circ}\).

    \(m \angle B=78^{\circ}\)

    Example \(\PageIndex{4}\)

    Using the following diagram, determine the measures of the missing angles.

    Solution

    First, state the relationship between the angle of \(\angle 1\) and \(\angle 3\).

    \(\angle 1\) and \(\angle 3\) are vertical angles and are equal in measure.

    Next, express the relationship using symbols.

    \(m\angle 1=m \angle 3\)

    Next, substitute the measure of \(\angle 1\) into the equation.

    \(m\angle 1=m \angle 3\)

    \(137^{\circ}=m\angle 3\)

    The answer is \(137^{\circ}\).

    \(m\angle 3= 137^{\circ}\)

    First, state the relationship between the angle of \(\angle 1\) and \(\angle 2\).

    \(\angle 1\) is adjacent to \(\angle 2\) and the two angles form a straight angle.

    Next, express the relationship using symbols.

    \(\angle 1+\angle 2=180^{\circ}\)

    Next, substitute the measure of \(\angle 1\) into the equation.

    \(137^{\circ}+\angle 2=180^{\circ}\)

    Next, subtract \(137^{\circ}\) from both sides of the equation.

    \(137^{\circ}+\angle 2=180^{\circ}\)

    \(137^{\circ}-137^{\circ}+\angle 2=180^{\circ}-137^{\circ}\)

    Then, simplify both sides of the equation.

    \(137^{\circ}-137^{\circ}+\angle 2=180^{\circ}-137^{\circ}\)

    \(\angle 2=43^{\circ}\)

    The answer is \(43^{\circ}\).

    \(m \angle 2=43^{\circ}\)

    First, state the relationship between the angle of \(\angle 2\) and \(\angle 4\).

    \(\angle 2\) and \(\angle 4\) are vertical angles and are equal in measure.

    Next, express the relationship using symbols.

    \(m \angle 2=m \angle 4\)

    Next, substitute the measure of \(\angle 2\) into the equation.

    \(m \angle 2=m \angle 4\)

    \(43^{\circ}=m \angle 4\)

    The answer is \(43^{\circ}\).

    \(m \angle 4=43^{\circ}\)

    Review

    If the following angle pairs are complementary, then what is the measure of the missing angle?

    1. If \(\angle A=45^{\circ}\) then \(\angle B=\)?

    2. If \(\angle C=83^{\circ}\) then \(\angle D=\)?

    3. If \(\angle E=33^{\circ}\) then \(\angle F=\)?

    4. If \(\angle G=53^{\circ}\) then \(\angle H=\)?

    If the following angle pairs are supplementary, then what is the measure of the missing angle?

    5. If \(\angle A=40^{\circ}\) then \(\angle B=\)?

    6. If \(\angle A=75^{\circ}\) then \(\angle B=\)?

    7. If \(\angle C=110^{\circ}\) then \(\angle F=\)?

    8. If \(\angle D=125^{\circ}\) then \(\angle E=\)?

    9. If \(\angle M=10^{\circ}\) then \(\angle N=\)?

    10. If \(\angle O=157^{\circ}\) then \(\angle P=\)?

    Define the following types of angle pairs.

    11. Vertical angles

    12. Adjacent angles

    13. Complementary angles

    14. Supplementary angles

    15. Interior angles

    Review (Answers)

    To see the Review answers, open this PDF file and look for section 6.4.

    Resources

    Vocabulary

    Term Definition
    Adjacent Angles Two angles are adjacent if they share a side and vertex. The word 'adjacent' means 'beside' or 'next-to'.
    Angle A geometric figure formed by two rays that connect at a single point or vertex.
    Intersecting lines Intersecting lines are lines that cross or meet at some point.
    Parallel Two or more lines are parallel when they lie in the same plane and never intersect. These lines will always have the same slope.
    Perpendicular lines Perpendicular lines are lines that intersect at a \(90^{\circ}\) angle.
    Straight angle A straight angle is a straight line equal to \(180^{\circ}\).

    Additional Resources

    Interactive Element

    Video: Complementary, Supplementary, and Vertical Angles

    Practice:Angle Properties and Theorems

    1.13: Angle Properties and Theorems (2024)

    FAQs

    What are the 12 angle theorems? ›

    The geometry theorems are: Isosceles Triangle Theorem, Angle Sum Triangle Theorem, Equilateral Triangle Theorem, Opposite Angle Theorem, Supplementary Angle Theorem, Complementary Angle Theorem, 3 Parallel Line Theorems, Exterior Angle Theorem, Exterior Angles of a Polygon and Interior Angles of a Polygon.

    What is angle sum theorem answers? ›

    The sum of the interior angles in a triangle is supplementary. In other words, the sum of the measure of the interior angles of a triangle equals 180°. So, the formula of the triangle sum theorem can be written as, for a triangle ABC, we have ∠A + ∠B + ∠C = 180°.

    What is the formula to solve an angle? ›

    The formula for Finding Angles
    Name of the FormulaFormula
    Sum of Interior angles Formula180°(n-2) Here, n is the number of sides of a polygon
    Trigonometric Ratiossin θ = opposite side/hypotenuse cos θ = adjacent side/hypotenuse tan θ = opposite side/adjacent side
    3 more rows
    Jul 13, 2022

    What is the theorem 1 in geometry? ›

    Theorem 1: If two lines intersect, then they intersect in exactly one point.

    What is the theorem 1 in math 10? ›

    Theorem 1:

    If a line is drawn parallel to one side of a triangle and intersects the other two sides, then the other two sides are divided in the same ratio. Construction: ABC is a triangle, DE is a line parallel to BC and intersecting AB at D and AC at E, i.e. DE || BC.

    What is the angle theorem 1? ›

    The angle at the centre ∠ A O B is 180°. Therefore, any inscribed angle ∠ A D B subtended by the same arc will be half the size of the centre angle (Theorem 1).

    What is the F rule for angles? ›

    F-Rule: Corresponding angles of parallel lines are equal. Of course, these angle relationships won't look like X's, U's, Z's and F's in all diagrams because the diagram may be rotated and consist of more lines.

    What are z angles called? ›

    These pairs of angles occur in a Z-shape, as indicated by the solid line in the diagram below. Such angles are called alternate angles. When a line intersects two parallel lines, alternate angles are equal.

    What is the Z rule for angles? ›

    Theorem 1 (The "Z" Theorem)

    If two lines are parallel then their alternate interior angles are equal. If the alternate interior angles of two lines are equal then the lines must be parallel.

    What is the formula for the angles theorem? ›

    That is, in a regular polygon with n edges, the equation for interior angles, a , is a = 180 ( n − 2 ) n . The formula can also be used to find a missing angle of a polygon. Since the theorem gives the sum of all interior angles, subtract the sum of the known angles from S n to obtain the missing angle.

    How do you solve right angle theorem? ›

    Proof of Right Angle Triangle Theorem
    1. Theorem:In a triangle, if square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.
    2. To prove: ∠B = 90°
    3. Proof: We have a Δ ABC in which AC2 = AB2 + BC2
    4. Also, read:
    5. c2 = a2 + b2
    6. c = √(a2 + b2)
    7. A = 1/2 b x h.

    What is the angle theorem rule? ›

    The exterior angle theorem states that when a triangle's side is extended, the resultant exterior angle formed is equal to the sum of the measures of the two opposite interior angles of the triangle. The theorem can be used to find the measure of an unknown angle in a triangle.

    How to solve for interior angle theorem? ›

    Interior Angle Formulas
    1. Method 1:
    2. Interior angles of a Regular Polygon = [180°(n) – 360°] / n.
    3. Method 2:
    4. Interior Angle of a polygon = 180° – Exterior angle of a polygon.
    5. Method 3:
    6. Interior Angle = Sum of the interior angles of a polygon / n.
    7. Statement:
    8. To prove:

    References

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