Question

Given: Circle O with diameter LN and inscribed angle LMN
Prove: Angle L M N is a right angle.
Circle O is shown. Line segment L N is a diameter. Points L, M, N, and K are on the circle. Lines connect each point.
What is the missing reason in step 5?
Statements
Reasons
1. circle O has diameter LN and inscribed angle LMN 1. given
2. Arc L K N is a semicircle 2. diameter Circle divides into 2 semicircles
3. circle O measures 360o 3.
measure of a circle is 360o
4. m Arc L K N = 180o 4. definition of semicircle
5. m∠LMN = 90o 5. ?
6. ∠LMN is a right angle 6. definition of right angle
HL theorem
inscribed angle theorem
diagonals of a rhombus are perpendicular.
formed by a tangent and a chord is half the measure of the intercepted arc.

Answer 1

To Prove :-

• ∠LMN is a right angle .

Answer :-

given that,

• O is the centre of the circle .
• LN is a diameter of the circle .

so,

→ LN is a straight line .

→ ∠LON = 180° { Straight line, Angle subtended by the arc LN at O .}

then,

→ ∠LMN = (1/2) ∠LON { Angle subtended by an arc at the center is double the angle subtended by it on any point on the remaining part of the circle. }

therefore,

→ ∠LMN = (1/2) * 180°

→ ∠LMN = 90° (Proved.)

Similarly,

→ ∠LKN = 90° .

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