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

    4.17: Triangle Angle Sum Theorem? ›

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

    What is theorem 4.1 triangle sum theorem? ›

    Theorem 4.1: Triangle Sum Theorem - The sum of the measures of the interior angles of a triangle is 180° Theorem 4.2: Exterior Angle Theorem - The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles.

    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 the formula for the angle sum property of a triangle? ›

    This property states that the sum of all the interior angles of a triangle is 180°. If the triangle is ∆ABC, the angle sum property formula is ∠A+∠B+∠C = 180°.

    What is the triangle sum theorem 9th grade? ›

    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 angle sum formula? ›

    The sum of the interior angles of a given polygon = (n − 2) × 180°, where n = the number of sides of the polygon.

    What is the theorem rule triangle? ›

    The angle sum property of the triangle states that A°+ B°+ C° = 180°. According to the Pythagoras Theorem, (AB)2 = (BC)2 + (CA)2. Triangle similarity theorems are used to prove the similarity between two triangles.

    What are the 4 triangle congruence theorems? ›

    SSS (Side-Side-Side) SAS (Side-Angle-Side) ASA (Angle-Side-Angle) AAS (Angle-Angle-Side)

    What is the theorem used to solve triangles? ›

    We can use the Pythagorean theorem and properties of sines, cosines, and tangents to solve the triangle, that is, to find unknown parts in terms of known parts. Pythagorean theorem: a2 + b2 = c2.

    What is the angle sum theorem? ›

    The triangle sum theorem, also known as the triangle angle sum theorem or angle sum theorem, is a mathematical statement about the three interior angles of a triangle. The theorem states that the sum of the three interior angles of any triangle will always add up to 180 degrees.

    What are the theorems for triangle angles? ›

    Theorems of Triangles

    Theorem 1: The total of the three interior angles in any triangle is 180 degrees. Theorem 2: When a triangle side is constructed, the exterior angle formed is equal to the sum of the interior opposite angles.

    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 of the triangle theorem? ›

    According to this theorem, for any triangle, the sum of lengths of two sides is always greater than the third side. In other words, this theorem specifies that the shortest distance between two distinct points is always a straight line.

    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.

    What does theorem 4 state? ›

    The opposite angles in a cyclic quadrilateral add up to In plain terms: If you look at the cyclic quadrilateral (a four sided shape where the four vertices touch the circumference of a circle) below, each vertex is a point on the circumference of the circle.

    References

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