Erasmus used Annabel's method
Erasmus does not justify the claim that the perpendicular bisector of the horizontal side divides the 130° into two equal parts. He could do this by showing that the pentagon is symmetrical so that the bisector of the vertical side passes through the opposite vertex.
He also needs to explain that the perpendicular bisector then divides the pentagon into two congruent quadrilaterals. Then he can apply the property that the sum of the angles in a quadrilateral sum to 360°.
His calculation method is correct but he did not finish his working out.
Erasmus's use of Annabel's method gives the correct measure of x = 75°.
Tomas used Carlos's method
Tomas makes a false assumption that all the exterior angles are congruent.
He did not notice that the pentagon is not regular. The exterior angles are all congruent only when the polygon is regular.
Tomas should calculate the size of the exterior angles for each of the known 130° interior angles first.
The angles on a line sum to 180°, so there are three exterior angles of 50°.
360 − 3 × 50 = 360 − 150 = 210°.
So, the two missing exterior angles are congruent and sum to 210°. Each is 210 ÷ 2 = 105°.
Then, since the angles on a line sum to 180°, x + 105 = 180. So x = 75°
Katerina used Brian's method
Katerina is correct that a trapezoid and triangle are formed by the horizontal line, but she does not fully explain her reasoning. It is not clear that the quadrilateral is a trapezoid, or that the trapezoid is isosceles.
She needs to show the base of the quadrilateral is parallel to the top to show that the quadrilateral is a trapezoid.
The horizontal side has at each end the same angle. The slant sides are the same length. So the line joining the ends of those slant sides is parallel to the top (trapezoid).
The trapezoid is isosceles because the slant sides are equal in length and joined to the top by congruent angles (symmetry). So both base angles can be labeled b.
She is correct that the triangle is isosceles because it has two congruent sides. So the two unknown angles in the triangle are congruent and can both be labeled a.
Katerina made a numerical error in stating a = 50°.
The angles in a triangle sum to 180°.
2a = 180 − 130 = 50°
She had forgotten to divide by two.
Katerina's next piece of reasoning is faulty.
It is not true that the consecutive angles in every quadrilateral sum to 180°. For example, it is not true that any two consecutive angles in a trapezoid always sum to180°.
In a trapezoid, the angles formed by a transversal crossing the parallel sides forms a pair of supplementary angles.
Supplementary angles sum to 180°.
So b = 180 − 130 = 50°
Katerina also needs to finish her solution by finding x = a + b = 25 + 50 = 75°
Megan uses Diane's method
Diana divided the pentagon into three triangles to calculate the measure of x. There is not enough detail to specify a method.
Megan uses faulty reasoning with Diane's trisection.
She makes a false assumption that the triangles are all isosceles.
Megan would need to give reasons to support the assertion that the triangles are isosceles, and there are none beyond surface appearance since they are not!
Diane's trisection method can lead to a correct solution. The sum of the angles in a triangle is 180°.
So the total angle sum of the pentagon is 3 × 180 = 540°.
This could be provide using the formula for the sum of the angles in a polygon with n sides, 180(n − 2).
The interior angles sum is 540 and there are three known angles of 130°.
So 2x = 540 − 3 × 130, and x = 75°.
In Q4, it is not expected that students will show that Megan's assumption is false. However, we supply a solution in case you want to work on this with students.
Assuming that the triangles are isosceles leads to a contradiction, showing that the assumption is false. (Proof by contradiction.)
Megan assumes the three triangles formed are all isosceles triangles with two congruent base angles of 65°. Suppose she is correct.
Each has base line of equal length, the base angles of equal measure, two sides of equal length, the apex angles must also be congruent to each other, and the triangles are thus congruent.
Each apex angle would be 130 ÷ 3 = 43⅓°.
Since the triangles are isosceles, and the angles in a triangle sum to 180°, the two base angles are (180 − 43⅓ ) ÷ 2 = 68⅓°.
x cannot be both 68⅓° and 65°. The assumption leads to a contradiction, and must be false.