Complete the two-column proof.
Given: <2 ≅ <4, M<2=110°
Proof: m<3 = 70°
m∠2 = m∠4 Definition of congruent angles
m∠4 = 110º b. ________
∠3 and ∠4 are a linear pair Definition of a linear pair (shown in diagram)
∠3 and ∠4 are supplementary Linear Pair postulate
m∠3 + m∠4 = 180º c. _______
m∠3 = 70º d. ________
a. Given
b. Substitution
c. Addition Property of Equality
d. Subtraction Property of Equality
To complete the two-column proof, we need to provide reasons for statements (b), (c), and (d).
b. To find the measure of angle ∠4, we use the definition of congruent angles. Since ∠2 ≅ ∠4 and the measure of ∠2 is given as 110°, we can conclude that ∠4 is also 110°.
c. Next, we use the definition of a linear pair. In the given diagram, ∠2 and ∠4 are adjacent angles that form a straight line. By definition, angles that form a straight line are called a linear pair. Therefore, ∠2 and ∠4 are a linear pair.
d. From the Linear Pair postulate, we know that if two angles form a linear pair, then they are supplementary. In other words, the sum of their measures equals 180°. Therefore, we can write the equation: m∠3 + m∠4 = 180°.
Since we already know that m∠4 = 110° (from statement b), we can substitute that value into the equation: m∠3 + 110° = 180°.
To solve for m∠3, we subtract 110° from both sides of the equation:
m∠3 = 180° - 110°
m∠3 = 70°.
So, statement (d) can be completed as follows:
m∠3 = 70°.
b. ∠4 ≅ ∠2 (Congruence of angles is symmetric)
c. m∠3 + 110º = 180º (Substitution property of equality)
d. m∠3 = 70º (Subtraction property of equality)