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Elementary Geometry for College St...

6th Edition
Daniel C. Alexander + 1 other
ISBN: 9781285195698

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BuyFindarrow_forward

Elementary Geometry for College St...

6th Edition
Daniel C. Alexander + 1 other
ISBN: 9781285195698
Textbook Problem
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In the proof below, provide the missing reasons.

Given: 1 and 2 are complementary
1 is acute
Prove: 2 is also acute
PROOF
Statements Reasons
1. 1 and 2 are complementary 1. ?
(1) 2. m 1 + m 2 = 90 2. ?
3. 1 is acute 3. ?
(3) 4. Where m 1 = x , 0 < x < 90 4. ?
(2) 5. x + m 2 = 90 5. ?
(5) 6. m 2 = 90 x 6. ?
(4) 7. x < 0 < 90 x 7. ?
(7) 8. 90 x < 90 < 180 x 8. ?
(7), (8) 9. 0 < 90 x < 90 9. ?
(6), (9) 10. 0 < m / 2 < 90 10. ?
(10) 11. 2 is acute 11. ?

To determine

To state:

The reasons for given statements.

Explanation

Given:

Two complementary angles 1 and 2. Angle 2 is acute angle.

Approach:

If the sum of two angles is 90° then the angles are known as complementary angles.

Calculation:

The reasons for given statements is explained in table below.

Statements Reasons
1. 1 and 2 are complementary 1. Given
2. m1+m2=90° 2. The sum of measures of complementary angles 1 and 2 is 90°.
3. 1 is acute 3. Given
4. Where m1=x, 0<x<90° 4. 1 and 2 is complementary. So, 1 is acute angle, then 0<m1<90°
5. x+m2=90° 5. Substitute x for m1 in m1+m2=90°
6. m2=90°x 6. Subtract x from both sides.
7. x<0<90°x 7. Substitution
8. 90°x<90°<180°x 8. Add 90° to each term of x<0<90°x
9. 0<90°x<90° 9. If 90x<90<180x, then 90°x<90°
10. 0<m/2<90° 10. Substitute m2 for 90°x
11. 2 is acute 11. If 0<m/2<90°, then it can be noticed that 2 is acute

Table 1

Conclusion:

The reasons for given statement is shown in Table 1

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