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For the following proof of the sine addition formula, determine the missing statements and reasons in the questions below.

Isosceles Triangle Height v2

In triangle ABC, let point D be the intersection of the altitude (from vertex B to side

$bar{AC}$ ) and side

$bar{AC}$ such that

$AD + DC = AC$ . Let

$m ang ABD = theta$ and

$m ang DBC = alpha$ . Let

$T$ denote the area of a triangle. Prove that

$sin(theta + alpha) = sin(theta)cos(alpha) + sin(alpha)cos(theta)$ .

$\ \ \ \ \ \ \ \ \ \ \ \ \mathbf{"Statement"} \ \ \ \ \ \ \ \ \ \ \ \$	$\mathbf{"Reason"}$
$1. theta + alpha = m ang ABC$	$1. "Angle Addition Postulate"$
$2. T_{Delta ABC} = 1/2 sin(m ang ABC) AB \ BC$	$2. "SAS formula for the area of a triangle"$
$3. T_{Delta ABC} = 1/2 sin(theta+alpha) AB \ BC$	$3. "Substitution Property of Equality"$
$4. T_{Delta ABC} = 1/2 AB \ BC sin(theta+alpha)$	$4. ""$
$5. T_{Delta ABD} = 1/2 sin(theta) AB \ BD$	$5. "SAS formula for the area of a triangle"$
$6. T_{Delta BDC} = 1/2 sin(alpha) BD \ BC$	$6. "SAS formula for the area of a triangle"$
$7. bar{BD} " is an altitude of " Delta ABC$	$7. ""$
$8. bar{BD} _\|_ bar{AC}$	$8. "Definition of an altitude"$
$9.$	$9. "Definition of perpendicular lines"$
$10. Delta ADB, \ Delta BDC " are right triangles"$	$10. ""$
$11. ( \ \ BD \ \ )/{BC} = cos(alpha)$	$11. ""$
$12. BD = BC cos(alpha)$	$12. "Multiplication Property of Equality"$
$13. ( \ \ BD \ \ )/{AB} = cos(theta)$	$13. ""$
$14. BD = AB cos(theta)$	$14. "Multiplication Property of Equality"$
$15. T_{Delta ABD} = 1/2 sin(theta) AB \ BC cos(alpha)$	$15. "Substitution Property of Equality"$
$16. T_{Delta ABD} = 1/2 AB \ BC sin(theta) cos(alpha)$	$16. "Commutative Property of Multiplication"$
$17. T_{Delta BDC} = 1/2 sin(alpha) AB cos(theta) BC$	$17. "Substitution Property of Equality"$
$18. T_{Delta BDC} = 1/2 AB \ BC sin(alpha) cos(theta)$	$18. "Commutative Property of Multiplication"$
$19.$	$19. "Area of a shape is equal to the sum of the"$ $\ \ " areas of its partitioned shapes"$
$20. 1/2 AB \ BC sin(theta+alpha) = 1/2 AB \ BC sin(theta) cos(alpha)$ $\ \ \ + 1/2 AB \ BC sin(alpha)cos(theta)$	$20. ""$
$21. 1/2 AB \ BC sin(theta+alpha) =$ $\ \ \ 1/2 AB \ BC (sin(theta) cos(alpha) + sin(alpha)cos(theta))$	$21. "Distribution Property of Equality"$
$22. sin(theta+alpha) = sin(theta)cos(alpha) + sin(alpha)cos(theta)$	$22. "Division Property of Equality"$

Grade 11 Trigonometry CCSS: HSF-TF.C.9

What is the missing reason in step 11?

Ratios of right triangles
SAS Congruence Theorem
Pythagorean Theorem
SAS formula for area of a triangle

Question Info

Trigonometry Question

Grade 11 Trigonometry CCSS: HSF-TF.C.9