The acute angles of a right triangle are complementary
m∠ YZV+m∠ YVZ =90^(∘)
Definition of Complementary Angles
m∠ YZV+m∠ X =90^(∘)
Substitution Property
∠ X and ∠ YZV are complementary.
Definition of Complementary Angles
Practice makes perfect
Let's look at the given figure.
To prove that ∠ X and ∠ YZV are complementary, we will construct a two-column proof. Let's start with stating the given information and the statement that we will prove as our first step.
Given:& △ XWV is isosceles; ZY⊥ YV
Prove:& ∠ X and ∠ YZV are complementary.
The Isosceles Triangle Theorem states that if two sides of a triangle are congruent, then the angles opposite those sides are congruent. Using the theorem, we can write the following step.
2. Isosceles Triangle Theorem
∠ X ≅ ∠ WVX
Looking at the figure, we see that ∠ WVX and ∠ YVZ are vertical angles. In this case, we will use the Vertical Angles Theorem to write the third step.
3. Vertical Angles Theorem
∠ WVX≅∠ YVZ
Next, we will use the Transitive Property between second and third step to write the next step.
4. Transitive Property
∠ X≅∠ YVZ
By the definition of congruent angles, two angles are congruent if and only if they have the same measure of angle.
5. Definition of Congruent Angles
m∠ X=m∠ YVZ
Given that ZY⊥ YV, we can write the following step.
6. Perpendicular lines form right angle
m∠ YVZ=90^(∘)
Using the definition of right a triangle, let's write the seventh step.
7. Definition of Right Triangle
△ ZVY is a right triangle.
We know that the acute angles of a right triangle are complementary.
8. The acute angles of
a right triangle are complementary
∠ YZV and ∠ YVZ are complementary.
By the definition of complementary angles, the sum of the measures of complementary angles is 90^(∘).
9. Definition of Complementary Angles
m∠ YZV+m∠ YVZ =90^(∘)
From the fifth step, we will substitute ∠ X for ∠ YVZ into the equation.
10. Substitution Property
m∠ YZV+m∠ X =90^(∘)
Finally, using the definition of complementary angles one more time, we can complete the proof.
11. Definition of Complementary Angles
∠ X and ∠ YZV are complementary.
Combining these steps, let's construct the two-column proof.
Statements
Reasons
△ XWV is isosceles; ZY⊥ YV
Given
∠ X ≅ ∠ WVX
Isosceles Triangle Theorem
∠ WVX≅∠ YVZ
Vertical Angles Theorem
∠ X≅∠ YVZ
Transitive Property
m∠ X=m∠ YVZ
Definition of Congruent Angles
m∠ YVZ=90^(∘)
Perpendicular lines form right angle
△ ZVY is a right triangle.
Definition of Right Triangle
∠ YZV and ∠ YVZ are complementary.
The acute angles of a right triangle are complementary
