Question

suppose that you cut off the angles of the triangle as shown in the diagram below. you lay the three angles so that the vertices are at the same point, and one of the angles is between and adjacent to the other two angles. which of the following statements have you proven? there are 360° in a circle. a straight angle has a measure of 180°. the sum of the measures of the angles of a triangle is 180°. corresponding angles have equal measures. question 2 2 pts part of a proof that base angles of an isosceles triangle are congruent is shown below. (overline{np}); (overline{nr}) bisects (angle mnp) (angle mcongangle p) match each step of the proof with the justification for the step. (overline{nm}congoverline{np}); (overline{nr}) bisects (angle mnp) (angle mnrcongangle pnr) (overline{nr}congoverline{nr}) therefore, (\triangle nmrcong\triangle npr) (angle mcongangle p)

Explanation:

Step 1: First problem analysis

When we cut off the angles of a triangle and lay them with vertices at the same point (one angle between and adjacent to the other two), we are essentially forming a straight - angle. A straight - angle is \(180^{\circ}\), and this is a visual proof that the sum of the interior angles of a triangle is \(180^{\circ}\).

• The statement “There are \(360^{\circ}\) in a circle” is about a circle, not related to the triangle - angle - sum.
• A straight - angle has a measure of \(180^{\circ}\), not \(100^{\circ}\).
• Corresponding angles having equal measures is related to parallel lines and transversals, not this triangle - angle - sum operation.

Step 2: Second problem (proof of base - angles of isosceles triangle)

• \(\overline{NM}\cong\overline{NP};\overline{NR}\) bisects \(\angle MNP\) is given (Given).
• \(\angle MNR\cong\angle PNR\) because of the definition of an angle bisector (Definition of angle bisector).
• \(\overline{NR}\cong\overline{NR}\) by the reflexive property of congruence (Reflexive property of congruence).
• \(\triangle NMR\cong\triangle NPR\) by the Side - Angle - Side (SAS) congruence criterion (SAS congruence criterion).
• \(\angle M\cong\angle P\) because corresponding parts of congruent triangles are congruent (CPCTC - Corresponding Parts of Congruent Triangles are Congruent).

Answer:

For the first question: The sum of the measures of the angles of a triangle is \(180^{\circ}\).
For the second question:

• \(\overline{NM}\cong\overline{NP};\overline{NR}\) bisects \(\angle MNP\): Given
• \(\angle MNR\cong\angle PNR\): Definition of angle bisector
• \(\overline{NR}\cong\overline{NR}\): Reflexive property of congruence
• \(\triangle NMR\cong\triangle NPR\): SAS congruence criterion
• \(\angle M\cong\angle P\): CPCTC