Sorry, is someone able to help with these 2 questions please? I have been at it for a while now but my teacher will not help help me with it. Thank you!
Find the missing angles in the Parallelogram
m < A = m < F =
m < B = m < G =
m < C = m < H =
m < D = m < I =
m < E =
-------------------------------------------------
Fill in the missing reasons in the following proof.
Statement | Reasons |
DFGH is a kite | |
DF≅FG
| 2 |
FH ⊥ DG
| 3 Property of a kite |
m ∠ DPF=90° m ∠ GPF=90° | 4 Perpendicular segments form right angles. |
∠DPF≅∠GPF
| 5 |
PF ≅ PF
| 6 |
△DPF≅△GPF
| 7 |
PD ≅GP | 8 |
There are a multitude of ways to find the missing angle measures in the diagram. I'll just show you my observations.
1) \(m\angle A=121^{\circ}\)
\(\angle A\) and the angle across from it form vertical angles. Thus, by the vertical angles theorem, they are congruent.
2) \(m\angle B=59^{\circ}\)
\(\angle A\) and \(\angle B\) form a linear pair, so the angles are supplementary by the linear pair theorem. If the angles are supplementary, then the sum of the measure of the angles is 180 degrees.
3) \(m\angle C=59^{\circ}\)
\(\angle B\) and \(\angle C\) together form vertical angles. As aforementioned, this means that the measure of both angles are equal.
4) \(m\angle D=55^{\circ}\)
It is given info that the figure is a parallelogram, which by definition is a quadrilateral with two pairs of opposite sides parallel. Plus, the unnamed 55 degree angle and \(\angle D\) can be classified as alternate interior angles. Since this is true, those angles are congruent.
5) \(m\angle E=4^{\circ}\)
The unnamed 121 degree angle, \(\angle D,\) and \(\angle E\) are all angles in a common triangle. The triangle sum theorem states that the sum of the measures of the interior angles of a triangle is 180 degrees. We can use this theorem to solve for the remaining angle:
\(121+m\angle D+m\angle E=180\) | Use the substitution property of equality to substitute in the known value for the measure of the angle D. |
\(121+55+m\angle E=180\) | Simplify the left hand side as much as possible. |
\(176+m\angle E=180\) | Subtract 176 from both sides to isolate angle E. |
\(m\angle E=4^{\circ}\) | |
I now want you to try to figure out the rest of the angle measures on your own now! See if you can do it.
Statements | Reasons |
DFGH is a kite | 1. Given |
\(\overline{DF}\cong\overline{FG}\) | 2. Definition of a kite (A kite is a quadrilateral with 2 pairs of adjacent congruent sides) |
\(\overline{FH}\perp\overline{DG}\) | 3. Property of a kite |
\(m\angle DPF=90^{\circ}\\ m\angle GPF=90^{\circ}\) | 4. Perpendicular segments form right angles |
\(\angle DPF\cong\angle GPF\) | 5. Right Angles Congruence Theorem (All rights angles are congruent) |
\(\overline{PF}\cong\overline{PF}\) | 6. Reflexive Property of Congruence (Any geometric figure is congruent to itself) |
\(\triangle DPF\cong\triangle GPF\) | 7. Hypotenuse Leg Triangle Congruence Theorem (Two rights triangles with corresponding hypotenuses and a select leg congruent are congruent triangles) |
\(\overline{PD}\cong\overline{GP}\) | 8. Corresponding Parts of Congruent Triangle are Congruent (sometimes abridged to CPCTC) |