Prove that ΔABC and ΔEDC are similar. triangles ABC and DEC where angles A and E are right angles, AC equals 4, AB equals 3, BC equals 5, DC equals 15, DE equals 9, and CE equals 12

15 over 4 equals 12 over 5 equals 9 over 3 shows the corresponding sides are proportional; therefore, ΔABC ~ ΔEDC by the SAS Similarity Postulate.

∠DCE is congruent to ∠BCA by the Vertical Angles Theorem and 15 over 5 equals 12 over 4 shows the corresponding sides are proportional; therefore, ΔABC ~ ΔEDC by the SAS Similarity Postulate.

∠E and ∠A are right angles; therefore, these angles are congruent since all right angles are congruent. 12 over 4 equals 9 over 3 shows the corresponding sides are proportional; therefore, ΔABC ~ ΔEDC by the SSS Similarity Postulate.

∠DCE is congruent to ∠CBA by the Vertical Angles Theorem and 15 over 5 equals 12 over 4 shows the corresponding sides are proportional; therefore, ΔABC ~ ΔEDC by the SSS Similarity Postulate.

Since shows two corresponding sides are proportional and based on the vertical angles theorem, therefore, we can prove that by the SAS Similarity Theorem. (Option B).

Recall:

In the diagram given,

We know that, based on the vertical angle theorem, <DCE and <BCA are congruent.

If we the two sides that the congruent angles lie in between are proportional, then we can prove that both triangles are congruent by the SAS similar theorem.

Thus:

(this shows proportionality)

Therefore, since shows two corresponding sides are proportional and based on the vertical angles theorem, therefore, we can prove that by the SAS Similarity Theorem. (Option B).

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p ÷ r = tan(x)

step-by-step explanation:

given that there is a figure below shows a right triangle: a right triangle is shown with hypotenuse equal to q units and the length of the legs equal to p units and r units.

graph is missing so i will create the graph as per given information.

the angle between the legs having lengths p units and q units is y degrees. the angle between the legs having lengths r units and q units is x degrees.

now we need to find about what is p ÷ r equal to.

so we can use trigonometric ratio tan(x)=[altitude]/[base] to find the value of p ÷ r, because p is the altitude and r is the base.

p ÷ r=tan(x)

hence final answer is p ÷ r=tan(x).