Look for vertical angles. Remember, the measure of a right angle is 90.
See solution.
Practice makes perfect
We need to prove one of the Right Angle Theorems, which states the following.
Perpendicular lines intersect to form four right angles.
To prove it, we will use the given figure.
Using the diagram above, we can rewrite the theorem as follows.
Given:& l ⊥ m
Prove:& ∠ 1, ∠ 2, ∠ 3, and ∠ 4 are right angles
Remember, the measure of a right angle is 90. Now we are ready to start the proof. We will write the proof by using a two-column proof.
Statements
Reasons
1.
l ⊥ m
1.
Given
2.
m∠ 1 = 90
2.
Since l ⊥ m
3.
m∠ 1 + m∠ 2=180
3.
Supplement Theorem
4.
90+m∠ 2=180
4.
Substitution Method
5.
m∠ 2 = 90
5.
Addition Property of Equality
6.
∠ 2 ≅ ∠ 3
6.
Vertical Angles Theorem
7.
m∠ 2 = m∠ 3
7.
Definition of Congruent Angles
8.
m∠ 3 = 90
8.
Substitution Method
9.
∠ 1 ≅ ∠ 4
9.
Vertical Angles Theorem
10.
m∠ 1 = m∠ 4
10.
Definition of Congruent Angles
11.
m∠ 4 = 90
11.
Substitution Method
Alternative Solution
Another Way
Here, we will prove this theorem without using two-column proof table. We will use the figure above. We know that the lines l and m are perpendicular. Therefore, the measure of ∠ 1 is 90^(∘). m∠ 1 = 90 ^(∘) ✓
Also, ∠ 1 and ∠ 2 are supplementary angles, so the sum of their measures equals 180^(∘).
From here by the Vertical Angles Theorem the measures of the angles ∠ 2 and ∠ 3 are similar. Moreover, by the Definition of Congruent Angles the measures of these angles are equal. m∠ 2 &= m ∠ 3
&⇓
90^(∘) &= m ∠ 3 ✓
By following the same steps we can find that ∠ 1 and ∠ 4 have the equal measures. Therefore, intersecting perpendicular lines forms four right angle, and this is what we wanted to prove.
