Question

(05.01 MC)
(PLS HELP ME IT IS URGENT !!)

If cos(w°) = sin(z°), which of the following statements is true?

triangle ABD and ECD in which angle D measures 90 degrees, angle A equals w degrees, angle B equals x degrees, angle E equals y degrees, and angle C equals z degrees

a
z = x and ΔABD ≅ ΔECD

b
w = z and ΔABD ≅ ΔECD

c
z = x and ΔABD ~ ΔECD

d
w = z and ΔABD ~ ΔECD

Answer 1

Answer:

$\text{Option c is correct answer.}$

Step-by-step explanation:

$\text{Since ABC is right triangle, angles}\ w \text{ and }x\text{ are complementary.}\\\therefore\ w^\text{o}+x^\text{o}=90^\text{o}\\\text{or, } w^\text{o}=90^\text{o}-x^\text{o}$

$\text{Now we are given:}\\\text{cos}(w)^\text{o}=\text{sin}(z)^\text{o}\\\text{or, }\text{cos}(90-x^\text{o})=\text{sin}(z)^\text{o}\\\text{or, }\text{sin}(x^\text{o})=\text{sin}(z^o)\\\text{or, }x^\text{o}=z^\text{o}$

$\text{This makes triangles ABD and ECD similar triangles. Because one pair of}\\\text{angles in both triangles are already right angles and another pair of angles}\ w\\\text{ and}\ x\text{ are obtained to be equal.}$

$\text{So option c is correct.}$

Answer 2

Answer:

C) z = x and ΔABD ~ ΔECD

Step-by-step explanation:

The right triangle ABD has point E on side AD and point C on side BD. Angle D is the right angle. Points E and C are connected with a line segment. Angle DAB is marked as w°, angle ABD is marked as x°, angle DEC is marked as y°, and angle ECD is marked as z°.

By applying the cosine and sine trigonometric ratios in triangle ABD, we can express:

$\cos(w^{\circ}) = \dfrac{AD}{AB}$

$\sin(w^{\circ}) = \dfrac{DB}{AB}$

$\cos(x^{\circ}) =\dfrac{DB}{AB}$

$\sin(x^{\circ}) = \dfrac{AD}{AB}$

So, cos(w°) = sin(x°) and sin(w°) = cos(x°).

Given that cos(w°) = sin(z°), it follows that sin(z°) = sin(x°). So, z = x.

Since right triangles ABD and ECD share a common angle at vertex D (90°), and the corresponding angles ABD and ECD are congruent (z = x), they are similar by Angle-Angle (AA) triangle similarity.