Question

identify two statements that contradict each other.
i. ∠m is an obtuse angle.
ii. ( mangle m + mangle p = 90 )
iii. ( 180 - mangle m = 25 )
iv. ( mangle p = 120 )

a. i and ii
b. i and iii
c. i and iv
d. iii and iv

Explanation:

Step1: Recall definitions

An obtuse angle has measure greater than \(90^\circ\) and less than \(180^\circ\). Let's analyze each statement:

Step2: Analyze Statement I

\(\angle M\) is obtuse, so \(90^\circ < m\angle M < 180^\circ\).

Step3: Analyze Statement II

\(m\angle M + m\angle P = 90^\circ\). Then \(m\angle M = 90^\circ - m\angle P\). Since angle measures are non - negative, \(m\angle M\leqslant90^\circ\), which contradicts the fact that \(m\angle M>90^\circ\) (from Statement I).

Step4: Analyze Statement III

Solve \(180 - m\angle M=25\) for \(m\angle M\). We get \(m\angle M = 180 - 25=155^\circ\), which is obtuse (consistent with Statement I).

Step5: Analyze Statement IV

\(m\angle P = 120^\circ\). There is no direct contradiction with Statement I as we don't know the relationship between \(\angle M\) and \(\angle P\) other than what is given in other statements. Also, Statement III and IV: from III, \(m\angle M = 155^\circ\), if we use II (\(m\angle M + m\angle P=90\)), it's a contradiction, but between III and IV, there's no inherent contradiction as we don't have a sum or other relation given between them.

Answer:

A. I and II