Question

name
2-3 reteach to build understanding
parallel lines and triangle angle sums

1. use the diagrams to fill in the blanks.

interior angle measures
diagram of triangle abc with angles 54° at a, 48° at c
m∠a + m∠b + m∠c = 180°
__ + m∠b + __ = 180°
m∠b + ____ = 180°
m∠b = ____
exterior angle measure
diagram of triangle abc with angle 59° at a, 40° at b, and dc parallel to ab?
m∠dca = m∠a + m∠b
m∠dca = __ + __
m∠dca = ____

1. from the figure shown, abby concludes that m∠1 = 85°. what is abby’s error? how would you correct the error?

diagram of a triangle with angles 62° and 33°, and a straight line with angles 1 and 2

1. complete the sentences to find the value of x.

the measure of the __ angle of the triangle equals the sum of the measures of its two __ angles.
diagram of a triangle with exterior angle 117° and interior angle 29°, and x°
so the __ angle equals the sum of the __ angle and the x° angle.
so, the x° angle is the ____ of the 117° angle and the 29° angle.
therefore, x = ____.

Explanation:

Problem 1 (Interior Angle Measures)

Step1: Substitute known angles

We know \( m\angle A = 54^\circ \) and \( m\angle C = 48^\circ \). So substitute into \( m\angle A + m\angle B + m\angle C = 180^\circ \):
\( 54^\circ + m\angle B + 48^\circ = 180^\circ \)

Step2: Combine like terms

Add \( 54^\circ \) and \( 48^\circ \):
\( m\angle B + 102^\circ = 180^\circ \)

Step3: Solve for \( m\angle B \)

Subtract \( 102^\circ \) from both sides:
\( m\angle B = 180^\circ - 102^\circ = 78^\circ \)

Problem 1 (Exterior Angle Measure)

Step1: Substitute known angles

We know \( m\angle A = 59^\circ \) and \( m\angle B = 40^\circ \). So substitute into \( m\angle DCA = m\angle A + m\angle B \):
\( m\angle DCA = 59^\circ + 40^\circ \)

Step2: Calculate the sum

Add \( 59^\circ \) and \( 40^\circ \):
\( m\angle DCA = 99^\circ \)

Problem 2

Abby likely used the wrong angle relationship. The triangle has angles \( 62^\circ \), \( 33^\circ \), so \( m\angle 2 = 180^\circ - 62^\circ - 33^\circ = 85^\circ \), but \( \angle 1 \) and \( \angle 2 \) are supplementary (form a linear pair), so \( m\angle 1 = 180^\circ - 85^\circ = 95^\circ \). Her error was not recognizing \( \angle 1 \) and \( \angle 2 \) are supplementary. To correct, find \( m\angle 2 \) first (via triangle angle sum) then subtract from \( 180^\circ \) to get \( m\angle 1 \).

Answer:

Abby’s error: She did not recognize \( \angle 1 \) and \( \angle 2 \) are supplementary (linear pair). She should calculate \( m\angle 2 = 180^\circ - 62^\circ - 33^\circ = 85^\circ \), then \( m\angle 1 = 180^\circ - 85^\circ = 95^\circ \).

Problem 3