Question

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

Drag an expression or phrase to each box to complete the proof.

Answer 1

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.

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

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

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.