4.17: Triangle Angle Sum Theorem (2024)

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    The interior angles of a triangle add to 180 degrees Use equations to find missing angle measures given the sum of 180 degrees.

    Triangle Sum Theorem

    The Triangle Sum Theorem says that the three interior angles of any triangle add up to \(180^{\circ}\).

    4.17: Triangle Angle Sum Theorem (1)

    \(m\angle 1+m\angle 2+m\angle 3=180^{\circ}\).

    Here is one proof of the Triangle Sum Theorem.

    4.17: Triangle Angle Sum Theorem (2)

    Given: \(\Delta ABC\) with \(\overleftrightarrow{AD} \parallel \overline{BC}\)

    Prove: \(m\angle 1+m\angle 2+m\angle 3=180^{\circ}\)

    Statement Reason
    1. \(\Delta ABC with \overleftrightarrow{AD} \parallel \overline{BC}\) Given
    2. \\(angle 1\cong \angle 4,\: \angle 2\cong \angle 5\) Alternate Interior Angles Theorem
    3. \(m\angle 1=m\angle 4,\: m\angle 2=m\angle 5\) \cong angles have = measures
    4. \(m\angle 4+m\angle CAD=180^{\circ}\) Linear Pair Postulate
    5. \(m\angle 3+m\angle 5=m\angle CAD\) Angle Addition Postulate
    6. \(m\angle 4+m\angle 3+m\angle 5=180^{\circ}\) Substitution PoE
    7. \(m\angle 1+m\angle 3+m\angle 2=180^{\circ}\) Substitution PoE

    You can use the Triangle Sum Theorem to find missing angles in triangles.

    What if you knew that two of the angles in a triangle measured \(55^{\circ}\)? How could you find the measure of the third angle?

    Example \(\PageIndex{1}\)

    Two interior angles of a triangle measure \(50^{\circ}\) and \(70^{\circ}\). What is the third interior angle of the triangle?

    Solution

    \(50^{\circ}+70^{\circ}+x=180^{\circ}\).

    Solve this equation and you find that the third angle is \(60^{\circ}\).

    Example \(\PageIndex{2}\)

    Find the value of \(x\) and the measure of each angle.

    4.17: Triangle Angle Sum Theorem (3)

    Solution

    All the angles add up to \(180^{\circ}\).

    \(\begin{align*} (8x−1)^{\circ}+(3x+9)^{\circ}+(3x+4)^{\circ}&=180^{\circ} \\ (14x+12)^{\circ}&=180^{\circ} \\ 14x&=168 \\ x&=12\end{align*} \)

    Substitute in 12 for \(x\) to find each angle.

    \([3(12)+9]^{\circ}=45^{\circ} \qquad [3(12)+4]^{\circ}=40^{\circ} \qquad [8(12)−1]^{\circ}=95^{\circ}\)

    Example \(\PageIndex{3}\)

    What is m\angle T?

    4.17: Triangle Angle Sum Theorem (4)

    Solution

    We know that the three angles in the triangle must add up to \(180^{\circ}\). To solve this problem, set up an equation and substitute in the information you know.

    \(\begin{align*} m\angle M+m\angle A+m\angle T&=180^{\circ} \\ 82^{\circ}+27^{\circ}+m\angle T&=180^{\circ} \\ 109^{\circ}+m\angle T&=180^{\circ} \\ m\angle T &=71^{\circ}\end{align*}\)

    Example \(\PageIndex{4}\)

    What is the measure of each angle in an equiangular triangle?

    4.17: Triangle Angle Sum Theorem (5)

    Solution

    To solve, remember that \(\Delta ABC\) is an equiangular triangle, so all three angles are equal. Write an equation.

    \(\begin{align*} m\angle A+m\angle B+m\angle C &=180^{\circ} \\ m\angle A+m\angle A+m\angle A&=180^{\circ} \qquad &Substitute,\: all\: angles\: are \: equal. \\ 3m\angle A&=180^{\circ} \qquad &Combine\:like \:terms. \\ m\angle A&=60^{\circ}\end{align*}\)

    If \(m\angle A=60^{\circ}\), then \(m\angle B=60^{\circ}\) and \(m\angle C=60^{\circ}\).

    Each angle in an equiangular triangle is \(60^{\circ}\).

    Example \(\PageIndex{5}\)

    Find the measure of the missing angle.

    4.17: Triangle Angle Sum Theorem (6)

    Solution

    We know that \(m\angle O=41^{\circ}\) and \(m\angle G=90^{\circ}\) because it is a right angle. Set up an equation like in Example 3.

    \(\begin{align*} m\angle D+m\angle O+m\angle G&=180^{\circ} \\ m\angle D+41^{\circ}+90^{\circ}&=180^{\circ} \\ m\angle D+41^{\circ}&=90^{\circ}\\ m\angle D=49^{\circ}\end{align*}\)

    Review

    Determine \(m\angle 1\) in each triangle.

    1.

    4.17: Triangle Angle Sum Theorem (7)

    2.

    4.17: Triangle Angle Sum Theorem (8)

    3.

    4.17: Triangle Angle Sum Theorem (9)

    4.

    4.17: Triangle Angle Sum Theorem (10)

    5.

    4.17: Triangle Angle Sum Theorem (11)

    6.

    4.17: Triangle Angle Sum Theorem (12)

    7.

    4.17: Triangle Angle Sum Theorem (13)

    8. Two interior angles of a triangle measure \(32^{\circ}\) and \(64^{\circ}\). What is the third interior angle of the triangle?

    9. Two interior angles of a triangle measure \(111^{\circ}\) and \(12^{\circ}\). What is the third interior angle of the triangle?

    10. Two interior angles of a triangle measure \(2^{\circ}\) and \(157^{\circ}\). What is the third interior angle of the triangle?

    Find the value of \(x\) and the measure of each angle.

    11.

    4.17: Triangle Angle Sum Theorem (14)

    12.

    4.17: Triangle Angle Sum Theorem (15)

    13.

    4.17: Triangle Angle Sum Theorem (16)

    14.

    4.17: Triangle Angle Sum Theorem (17)

    15.

    4.17: Triangle Angle Sum Theorem (18)

    Review (Answers)

    To see the Review answers, open this PDF file and look for section 4.1.

    Resources

    Vocabulary

    Term Definition
    Triangle Sum Theorem The Triangle Sum Theorem states that the three interior angles of any triangle add up to 180 degrees.

    Additional Resources

    Interactive Element

    Video: Triangle Sum Theorem Principles - Basic

    Activities: Triangle Sum Theorem Discussion Questions

    Study Aids: Triangle Relationships Study Guide

    Practice: Triangle Angle Sum Theorem

    Real World: Triangle Sum Theorem

    4.17: Triangle Angle Sum Theorem (2024)

    FAQs

    What is the answer to the triangle sum theorem? ›

    Answer: The sum of the three angles of a triangle is always 180 degrees. To find the measure of the third angle, find the sum of the other two angles and subtract that sum from 180.

    What is the 4.11 theorem? ›

    Third angles are equal if the other two sets are each congruent. If two angles in one triangle are congruent to two angles in another triangle, then the third pair of angles must also congruent. This is called the Third Angle Theorem.

    What is the theorem 4 1 in geometry? ›

    Theorem 4-1 Congruence of angles is reflexive, symmetric, and transitive. Theorem 4-2 If two angles are supplementary to then same angle, the they are congruent. other. Theorem 4-4 If two angles are complementary to the same angle, then they are congruent to each other.

    What is the theorem 4 of triangles? ›

    Theorem 4: If in two triangles, the sides of one triangle are proportional to the sides of the other triangle, then their corresponding angles are equal and hence the two triangles are similar.

    What is theorem 4-3 isosceles triangle theorem? ›

    Theorem 4-3 (Isosceles Triangle Thm): If two sides of a triangle are congruent, then the angles opposite those sides are congruent.

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