Question

1. in parallelogram mnpq, m∠n = 6x + 10° and m∠m = 5x + 18.5°. how many degrees is ∠m?
2. match the following diagrams with their descriptions:

26 - 1. median
26 - 2. angle bisector
26 - 3. perpendicular bisector

1. match the following centers with their intersections on a triangle:

27 - 1. incenter
27 - 2. circumcenter
27 - 3. centroid
a. angle bisectors
b. medians
c. perpendicular bisectors

1. in the diagram below, ef is the median of trapezoid abcd. if ab = 4, dc = x - 3, and ef = 2x+2, what is the value of x?

Explanation:

Step1: Use parallelogram property

In a parallelogram, opposite - angles are equal. So, if we assume \(\angle Y\) and \(\angle M\) are opposite angles, then \(6x + 10=5x + 18.5\).

Step2: Solve the equation for \(x\)

Subtract \(5x\) from both sides: \(6x-5x + 10=5x-5x + 18.5\), which simplifies to \(x+10 = 18.5\). Then subtract 10 from both sides: \(x=18.5 - 10\), so \(x = 8.5\).

Step3: Find the measure of \(\angle Y\)

Substitute \(x = 8.5\) into the expression for \(\angle Y\): \(m\angle Y=6x + 10=6\times8.5+10=51 + 10=61^{\circ}\).

26 - 1: A (A median of a triangle is a line - segment that joins a vertex to the mid - point of the opposite side, and in figure A, \(CF\) is joining vertex \(C\) to the mid - point of \(AB\)).
26 - 2: B (An angle bisector divides an angle into two equal parts, and in figure B, \(CD\) is dividing \(\angle ACB\) into two equal parts).
26 - 3: C (A perpendicular bisector is a line that is perpendicular to a line - segment and also bisects it. In figure C, \(CE\) is perpendicular to \(BD\) and bisects it).

27 - 1: A (The in - center of a triangle is the point of intersection of the angle bisectors of the triangle).
27 - 2: C (The circum - center of a triangle is the point of intersection of the perpendicular bisectors of the sides of the triangle).
27 - 3: B (The centroid of a triangle is the point of intersection of the medians of the triangle).

Step4: Use trapezoid median formula

The formula for the median of a trapezoid is \(EF=\frac{AB + DC}{2}\). Given \(AB=x + 9\), \(DC=x - 3\), and \(EF = 2x+2\). Substitute into the formula: \(2x+2=\frac{(x + 9)+(x - 3)}{2}\).

Step5: Simplify the right - hand side of the equation

First, simplify the numerator of the right - hand side: \((x + 9)+(x - 3)=x+x+9 - 3=2x + 6\). So the equation becomes \(2x+2=\frac{2x + 6}{2}\), and \(\frac{2x + 6}{2}=x + 3\).

Step6: Solve the equation for \(x\)

We have \(2x+2=x + 3\). Subtract \(x\) from both sides: \(2x-x+2=x-x + 3\), which gives \(x+2 = 3\). Then subtract 2 from both sides: \(x=3 - 2=1\).

Answer:

The measure of \(\angle Y\) is \(61^{\circ}\).
26 - 1: A
26 - 2: B
26 - 3: C
27 - 1: A
27 - 2: C
27 - 3: B
The value of \(x\) for the trapezoid is \(1\).