Question

identify the two statements that contradict each other in the set:
i. ( mangle k + mangle l = 110 )
ii. ( mangle k - mangle l = 20 )
iii. ( mangle k = 180 )
the two statements that contradict each other are

Explanation:

Step1: Analyze Statement I

Statement I: \( m\angle K + m\angle L = 150 \)

Step2: Analyze Statement II

Statement II: \( m\angle K - m\angle L = 20 \)
Let \( m\angle K = x \) and \( m\angle L = y \). Then we have the system of equations:

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Solving this system, we add the two equations: \( 2x = 170 \Rightarrow x = 85 \), then \( y = 150 - 85 = 65 \). So Statements I and II are consistent.

Step3: Analyze Statement III

Statement III: \( m\angle K = 180 \)
If \( m\angle K = 180 \), then from Statement I, \( 180 + m\angle L = 150 \Rightarrow m\angle L = -30 \), which is impossible because angle measures are non - negative. Also, from Statement II, \( 180 - m\angle L = 20 \Rightarrow m\angle L = 160 \), and then from Statement I, \( 180+160 = 340
eq150 \). So Statement III contradicts both Statement I and Statement II. But we need to find two statements that contradict each other. Let's check the sum and the value of \( m\angle K \). If \( m\angle K = 180 \) (Statement III), and from Statement I \( m\angle K+m\angle L = 150 \), since \( m\angle K = 180>150 \), this is a contradiction. Also, if we consider the system from I and II, we get a valid angle measure for \( \angle K \) (85), which is not 180. So the two statements that contradict each other are I and III (or II and III, but I and III is more direct from the sum).

Answer:

I and III (or the statements: I. \( m\angle K + m\angle L=150 \) and III. \( m\angle K = 180 \))