Question

proving the congruent supplements theorem
given: ∠1 and ∠2 are supplements, and ∠3 and ∠2 are supplements.
prove: ∠1 ≅ ∠3
complete the missing parts of the paragraph proof.
by the definition of supplementary angles, the sum of the measures of angles 1 and 2 is 180 degrees. likewise, the sum of the measures of angles blank is 180 degrees. by the blank property, m∠1 + m∠2 = m∠3 + m∠2. subtract the measure of angle blank from each side. you get m∠1 = m∠3, or ∠1 ≅ ∠3, by the definition of congruence.
image of angles 1, 2, 3

Explanation:

Step1: Identify the second pair of supplementary angles

Given that ∠3 and ∠2 are supplements, so the sum of ∠3 and ∠2 is 180 degrees. So the first missing blank is "3 and 2".

Step2: Identify the property used

The equation \( m\angle1 + m\angle2 = m\angle3 + m\angle2 \) is derived from the fact that both sums equal 180 (from supplementary angles) and this is an application of the "transitive" property (if \( a = c \) and \( b = c \), then \( a = b \)) or more accurately the "substitution" or "equality" property, but in the context of angle measures and supplementary angles, the "transitive" or "equality" (specifically the "substitution" or "addition" but here it's about equal sums leading to equal equations) - actually, the property that if two quantities are equal to the same quantity, they are equal to each other is the "transitive property of equality", but more commonly, when we set the two sums equal (since both equal 180), we use the "substitution" or "equality" property. However, the key here is that the equation \( m\angle1 + m\angle2 = m\angle3 + m\angle2 \) is from the fact that both \( m\angle1 + m\angle2 = 180 \) and \( m\angle3 + m\angle2 = 180 \), so by the "transitive property of equality" (or "substitution"), we can set them equal. So the second missing blank is "transitive" (or "equality", but in the context of angle proofs, "transitive" or "substitution" - however, the standard property here is the "substitution" or "equality" property, but the most appropriate is "transitive" or "equality". Wait, actually, the property that if \( a = c \) and \( b = c \), then \( a = b \) is the transitive property of equality. So we use the "transitive" property here.

Step3: Identify the angle to subtract

To solve \( m\angle1 + m\angle2 = m\angle3 + m\angle2 \) for \( m\angle1 \) and \( m\angle3 \), we subtract \( m\angle2 \) from both sides. So the third missing blank is "2".

Answer:

First blank: 3 and 2; Second blank: transitive (or equality); Third blank: 2

(Note: The exact property name might vary slightly in different curricula, but the key steps are as above. The first blank is "3 and 2", the second is a property related to equality (like transitive or substitution), and the third is "2".)