Chapter 4: Q17. (page 138)

Write proofs in two-column form.
Given: $\bar{XY} ≅ \bar{XZ}; \bar{OY} ≅ \bar{OZ}$
Prove: $m ∠ 1 ≅ m ∠ 4$

Short Answer

Expert verified

Statement	Reason
$\bar{XY} ≅ \bar{XZ}$	Given
$m ∠ 1 + m ∠ 2 = m ∠ 3 + m ∠ 4$	Isosceles triangle theorem
$\bar{OY} ≅ \bar{OZ}$	Given
$m ∠ 2 = m ∠ 3$	Isosceles triangle theorem
$m ∠ 1 = m ∠ 4$	Since , so cancels out
$m ∠ 1 = m ∠ 4$	Proved

Step by step solution

Step 1. Apply isosceles triangle theorem to ΔXYZ.

If the two sides of a triangle are congruent, then the angles opposite to those sides are congruent.

It is given that, $\bar{X Y} ≅ \bar{X Z}$ therefore by isosceles triangle theorem $∠ X Y Z ≅ ∠ X Z Y$ .

But $m ∠ X Y Z = m ∠ 1 + m ∠ 2$ and $m ∠ X Z Y = m ∠ 3 + m ∠ 4$ .

Then $m ∠ 1 + m ∠ 2 = m ∠ 3 + m ∠ 4$

Step 2. Apply isosceles triangle theorem to ΔOYZ.

It is given that, $\bar{O Y} ≅ \bar{O Z}$ , therefore by isosceles triangle theorem, $∠ 2 ≅ ∠ 3$ , that is $m ∠ 2 = m ∠ 3$ .

Step 3. Description of step.

Substitute $m ∠ 2$ for $m ∠ 3$ into $m ∠ 1 + m ∠ 2 = m ∠ 3 + m ∠ 4$ and simplify.

$\begin{matrix} m ∠ 1 + m ∠ 2 = m ∠ 2 + m ∠ 4 \\ m ∠ 1 = m ∠ 4 \end{matrix}$

Hence it is proved that $m ∠ 1 = m ∠ 4$ .

Step 4. Two column proof.

Write a two-column proof based on above explanation.

Statement	Reason
$\bar{X Y} ≅ \bar{X Z}$	Given
$m ∠ 1 + m ∠ 2 = m ∠ 3 + m ∠ 4$	Isosceles triangle theorem
$\bar{O Y} ≅ \bar{O Z}$	Given
$m ∠ 2 = m ∠ 3$	Isosceles triangle theorem
$m ∠ 1 = m ∠ 4$	Since , so cancels out
$m ∠ 1 = m ∠ 4$	Proved

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Write proofs in two-column form.
Given: $\bar{XY} ≅ \bar{XZ}; \bar{OY} ≅ \bar{OZ}$
Prove: $m ∠ 1 ≅ m ∠ 4$

Short Answer

Step by step solution

Step 1. Apply isosceles triangle theorem to ΔXYZ.

Step 2. Apply isosceles triangle theorem to ΔOYZ.

Step 3. Description of step.

Step 4. Two column proof.

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Write proofs in two-column form.Given: XY¯≅XZ¯;OY¯≅OZ¯Prove:m∠1≅m∠4

Short Answer

Step by step solution

Step 1. Apply isosceles triangle theorem to ΔXYZ.

Step 2. Apply isosceles triangle theorem to ΔOYZ.

Step 3. Description of step.

Step 4. Two column proof.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Probability and Statistics

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Study anywhere. Anytime. Across all devices.

Write proofs in two-column form.
Given: $\bar{XY} ≅ \bar{XZ}; \bar{OY} ≅ \bar{OZ}$
Prove: $m ∠ 1 ≅ m ∠ 4$