Question

A conjecture and the flowchart proof used to prove the conjecture are shown.

Answer 1

Answer:

1. (Box on the left): Given.

2. (Box on the right): Angle Addition Postulate.

3. $51$°+$39$°=$m$∠DEF

4. Definition of right triangle.

5. ∠DEF is a right triangle.

Step-by-step explanation:

1. You have the angles given in the first figure.

2 and 3. The Angle Addition Postulate establishes that if GE is in the interior of the angle ∠DEF, then:

$51$°+$39$°=$m$∠DEF

4. A rigth triangle is a triangle that has a angle of 90 degrees.

5. The angle ∠DEF measures 90 degrees, therefore, the triangle is a right triangle.

Answer 2

Answer:

Blank 1: Given

Blank 2: Angle addition postulate

Blank 3: $51^{\circ}+39^{\circ}=m\angle DE F$

Blank 4: $90^{\circ}=m\angle DE F$

Blank 5: ∠DEF is a right angle.

Blank 6: Definition of right triangle

Step-by-step explanation:

Given: $m\angle DEG=51^{\circ}$ and $m\angle GEF=39^{\circ}$.

Prove: ΔDEF is a right triangle.

Proof:

$m\angle DEG=51^{\circ},m\angle GEF=39^{\circ}$      (Given)

$m\angle DEG+m\angle GEF=m\angle DE F$         (Angle addition postulate)

Substitute $m\angle DEG=51^{\circ}$ and $m\angle GEF=39^{\circ}$ in the above equation.

$51^{\circ}+39^{\circ}=m\angle DE F$         (Substitution property of equality)

$90^{\circ}=m\angle DE F$                  (On simplifying)

∠DEF is a right angle.                                    (Definition of right angle)

ΔDEF is a right triangle                        (Definition of right triangle)

Hence proved.