7. In the diagram, LMN = ONM = 90°. P is the midpoint of MN,
MN = 2ML and MN = NO. Prove that:
a) the triangles MNL and NOP are congruent
b) OÊN = LÑO
c) LỘO = 90°
Answers
Answer:
In the diagram,LMN = ONM= 90°. P is the midpoint of MN, MN = 2ML and
MN = NO
Step-by-step explanation:
Step 1
(A) Angle LMN = angle ONM [given]
MN = NO [given]
P is the midpoint of MN [given]
MN = 2 PN [definition of midpoint]
MN = 2 ML [given]
2 ML = 2 PN [transitivity of equality]
ML = PN [division property of equality]
Triangles MNL and NOP are congruent [side-angle-side]
Step 2
(B) Angles LMN and ONM measure 90° [given]
Measures of angles LMN, MNL, and NLM sum to 180° [angles of a triangle]
Measures of angles MNL and NLM sum to 90° [subtraction property of equality]
Measures of angles MNL and LNO sum to 90° [adjacent angles]
Measures of angles MNL and NLM = measures of angles MNL and LNO [transitive property of equality]
Measures of angles NLM and LNO are equal [subtraction property of equality]
Angles NLM and LNO are congruent [definition of congruent angles]
Triangles MNL and NOP are congruent [proved in part (a)]
Angles OPN and NLM are congruent [corresponding angles of congruent triangles]
Angles OPN and LNO are congruent [transitive property of angle congruence]
(C) Angles QNO and LNO are congruent [same angle]
Angles NOQ and NOP are congruent [same angle]
Angles OPN and LNO are congruent [proved in part (b)]
Angles QNO and OPN are congruent [transitive property of angle congruence]
Triangles NOP and QON are similar [angle-angle similarity]
Angles NQO and PNO have the same measure [corresponding angles of similar triangles]
Angle MNO measures 90° [given]
Angle PNO measures 90° [same angle]
Angle NQO measures 90° [equal angles]
LN is perpendicular to PO [definition of perpendicular]
Angle LQO measures 90° [definition of perpendicular]
Thus we can say that LQO=90°
Step-by-step explanation:
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