The Triangle Angle Sum Theorem states that the sum of the interior angles of a triangle is always 180 degrees. The three angles in a triangle are typically labeled as \(A\), \(B\), and \(C\). Thus, we have the relation:
\[ A + B + C = 180^\circ \]
Here are a few practice problems based on the Triangle Angle Sum Theorem:
Problem 1:
In triangle ABC, angle A is 50 degrees and angle B is 70 degrees. What is the measure of angle C?
Solution:
Using the Triangle Angle Sum Theorem:
\[ A + B + C = 180^\circ \] \[ 50^\circ + 70^\circ + C = 180^\circ \]
Combining the known angles:
\[ 120^\circ + C = 180^\circ \]
Subtracting 120 degrees from both sides:
\[ C = 180^\circ - 120^\circ = 60^\circ \]
Problem 2:
In triangle DEF, angle D measures 35 degrees and angle E measures 95 degrees. Find the measure of angle F.
Solution:
Using the Triangle Angle Sum Theorem:
\[ D + E + F = 180^\circ \] \[ 35^\circ + 95^\circ + F = 180^\circ \]
Combining the known angles:
\[ 130^\circ + F = 180^\circ \]
Subtracting 130 degrees from both sides:
\[ F = 180^\circ - 130^\circ = 50^\circ \]
Problem 3:
In triangle GHI, angle G is three times angle H, and angle I is 20 degrees. Find the measures of angles G and H.
Solution:
Let the measure of angle H be \(x\). Then, angle G will be \(3x\). According to the Triangle Angle Sum Theorem:
\[ G + H + I = 180^\circ \] \[ 3x + x + 20^\circ = 180^\circ \] \[ 4x + 20^\circ = 180^\circ \]
Subtracting 20 degrees from both sides:
\[ 4x = 160^\circ \]
Dividing by 4:
\[ x = 40^\circ \]
So, angle H is \(40^\circ\), and angle G is:
\[ G = 3x = 3(40^\circ) = 120^\circ \]
Problem 4:
In triangle JKL, angle J is twice angle K, and angle L is 40 degrees. Find the measures of angles J and K.
Solution:
Let the measure of angle K be \(y\). Then, angle J will be \(2y\). According to the Triangle Angle Sum Theorem:
\[ J + K + L = 180^\circ \] \[ 2y + y + 40^\circ = 180^\circ \] \[ 3y + 40^\circ = 180^\circ \]
Subtracting 40 degrees from both sides:
\[ 3y = 140^\circ \]
Dividing by 3:
\[ y = \frac{140^\circ}{3} \approx 46.67^\circ \]
So, angle K measures approximately \(46.67^\circ\), and angle J is:
\[ J = 2y = 2 \left(\frac{140^\circ}{3}\right) = \frac{280^\circ}{3} \approx 93.33^\circ \]
Summary
Using the Triangle Angle Sum Theorem allows you to find unknown angles in a triangle when given some known angles. You can practice with different angle measurements or relationships between angles for further understanding!
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