Given: Circle O with diameter LN and inscribed angle LMN
Prove: $\angle LMN$ is a right angle.
What is the missing reason in step 5?
| | Statements | | Reasons |
| :--------- | :---------------------------------------------------- | :--------- | :--------------------------------------------------------- |
| 1. | circle O has diameter LN and inscribed angle LMN | 1. | given |
| 2. | $\overleftarrow{ KN }$ is a semicircle | 2. | diameter © divides into 2 semicircles |
| 3. | circle O measures $360^{\circ}$ | 3. | measure of a circle is $360^{\circ}$ |
| 4. | $m \overline{ LKN }=180^{\circ}$ | 4. | definition of semicircle |
| 5. | $m \angle LMN =90^{\circ}$ | 5. | ? |
| 6. | $\angle LMN$ is a right angle | 6. | definition of right angle |
A. HL theorem
B. inscribed angle theorem
C. diagonals of a rhombus are perpendicular.
Answer (2)
The problem provides a geometric proof concerning a circle, diameter, and inscribed angle.
The inscribed angle theorem states that an inscribed angle's measure is half its intercepted arc's measure.
Since the intercepted arc is a semicircle ( 18 0 ∘ ), the inscribed angle measures 9 0 ∘ .
Therefore, the missing reason is the inscribed angle theorem: in scr ib e d an g l e t h eore m .
Explanation
Problem Analysis We are given a circle O with diameter LN and an inscribed angle LMN. We want to prove that angle LMN is a right angle. The proof table provides a step-by-step approach, and our task is to identify the missing reason in step 5.
Inscribed Angle Theorem The inscribed angle theorem states that the measure of an inscribed angle is half the measure of its intercepted arc. In this case, the inscribed angle LMN intercepts arc LKN.
Applying the Theorem Step 4 states that the measure of arc LKN is 180 degrees ( m L K N = 18 0 ∘ ). According to the inscribed angle theorem, the measure of angle LMN is half the measure of its intercepted arc LKN. Therefore, m ∠ L MN = 2 1 m L K N = 2 1 ( 18 0 ∘ ) = 9 0 ∘ .
Conclusion Thus, the missing reason in step 5 is the inscribed angle theorem.
Examples
Imagine you're designing a circular pizza and want to cut a slice that's exactly a right angle. If you ensure the slice's corners touch the edge of the pizza and the slice spans half the pizza (a semicircle), you've created a perfect right angle slice! This is a direct application of the inscribed angle theorem, ensuring your pizza slice is geometrically sound and delicious.
The missing reason in step 5 is the inscribed angle theorem, which states that an inscribed angle is half the measure of its intercepted arc. Since the intercepted arc L K N measures 18 0 ∘ , the angle ∠ L MN is calculated to be 9 0 ∘ . Therefore, ∠ L MN is a right angle.
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