Question

Given the figure above choose the correct statements if ∠=84°m∠DCE=84°, and ∠=62°m∠EBF=62° then

Answer 1

If m∠DCE = 84°, and m∠EBF = 62°, then the correct statements include;

a. m∠DEF = 34°.

b. m∠EDF = 62°

d. m∠DFA = 62°

e. m∠EDA = 96°

f. m∠DEB = 118°.

In Mathematics and Geometry, the triangle midpoint theorem states that the line segment which joins the midpoints of two (2) sides of a triangle is parallel to the third side, and it's congruent to one-half of the third side.

Based on the diagram, points D, E, and F are the midpoint of sides AC, BC, and AB respectively. By the midpoint theorem, we have the following parallel sides;

AC ║ EF

DE ║ AB

BC ║ DF

So, quadrilaterals DEFB and DAFE are parallelograms with congruent opposite (adjacent) angles;

m∠DAF ≅ m∠DEF

m∠EDF ≅ m∠EFB = 62°

By the triangle sum property, we have;

m∠DAF + m∠EBF + m∠DCE = 180°

m∠DAF = 180° - (84° + 62°)

m∠DAF ≅ m∠DEF = 34°.

Based on the corresponding angles theorem, we have these congruent angles;

m∠DFA ≅ m∠CED ≅ m∠EBF = 62°.

m∠DAF ≅ m∠CDE ≅ m∠EFB = 34°.

In triangle ADF, we have:

m∠DAF + m∠ADF + m∠DFA = 180°

m∠ADF = 180° - (34° + 62°)

m∠ADF = 84°.

Based on the consecutive interior angles theorem, we have;

m∠DEB + m∠EBF = 180°

m∠DEB = 180° - 62°

m∠DEB = 118°

Based on the linear pairs theorem, we have;

m∠CDE + m∠EDA = 180°

m∠EDA = 180° - 34°

m∠EDA = 146°

Answer 2

Given:

There is a triangle given with some data with two angles C=84 and B=62

Required:

We need to find that which atatement is right

Step-by-step explanation:

First find angle C

`\begin{gathered} \angle A+\angle B+\angle C=180\degree \\ 62+\angle A+84=180 \\ \angle A=34 \end{gathered}`

now by corresponding angles property

by this

`\begin{gathered} \angle CDE=\angle A=\angle EFB=34 \\ \angle CED=\angle B=\angle DFA=62 \\ \angle FDA=\angle C=\angle FEB=84 \end{gathered}`

the whole D is supplimentary angle so

`\begin{gathered} 34+\angle FDE+84=180 \\ \angle FDE=62=\angle B \end{gathered}`

Opposite angle

similarly

`\begin{gathered} \angle FED=\angle A=34 \\ and\text{ } \\ \angle DFE=\angle C=84 \end{gathered}`

Final answer:

Corrects are