Question

given : ( mangle acb = 110^circ ), ( mangle cde = 40^circ ), ( mangle b = 30^circ )
prove : ( \triangle eab sim \triangle dcb )

statement	reason
( mangle cde = 40^circ )
( mangle b = 30^circ )
( mangle acb + mangle a + mangle b = 180^circ )
( 110^circ + mangle a + 30^circ = 180^circ )
( mangle a + 140 = 180^circ )
( mangle a = 40^circ )
( mangle a = mangle cde )
( angle a cong angle cde )
( angle acb cong angle dce )
( \triangle acb sim \triangle dce )

options: given, triangle sum theorem, reflexive property, aa similarity, subtraction property, simplify, substitution property, definition of congruence

Explanation:

Step1: Fill in "Given" for angle measures from problem

For \( m\angle ACB = 110^\circ \), \( m\angle CDE = 40^\circ \), \( m\angle B = 30^\circ \), the reason is "Given" (as these are provided in the problem statement).

Step2: Triangle Sum Theorem for angle sum

The statement \( m\angle ACB + m\angle A + m\angle B = 180^\circ \) uses the "Triangle Sum Theorem" (sum of angles in a triangle is \( 180^\circ \)).

Step3: Substitution Property for angle values

Substitute \( m\angle ACB = 110^\circ \) and \( m\angle B = 30^\circ \) into the sum: \( 110^\circ + m\angle A + 30^\circ = 180^\circ \) (reason: "Substitution Property").

Step4: Simplify the equation

Simplify \( 110^\circ + 30^\circ \) to get \( m\angle A + 140^\circ = 180^\circ \) (reason: "Simplify").

Step5: Subtraction Property to solve for \( m\angle A \)

Subtract \( 140^\circ \) from both sides: \( m\angle A = 40^\circ \) (reason: "Subtraction Property").

Step6: Substitution to compare angles

Since \( m\angle A = 40^\circ \) and \( m\angle CDE = 40^\circ \), \( m\angle A = m\angle CDE \) (reason: "Substitution Property").

Step7: Definition of Congruence

If \( m\angle A = m\angle CDE \), then \( \angle A \cong \angle CDE \) (reason: "Definition of Congruence").

Step8: Reflexive Property for \( \angle ACB \) and \( \angle DCE \)

\( \angle ACB \cong \angle DCE \) because they are the same angle (reason: "Reflexive Property").

Step9: AA Similarity for triangles

With \( \angle A \cong \angle CDE \) and \( \angle ACB \cong \angle DCE \), \( \triangle ACB \sim \triangle DCE \) (reason: "AA Similarity").

(Note: To prove \( \triangle EAB \sim \triangle DCB \), we can extend the reasoning: \( \angle B \) is common to both \( \triangle EAB \) and \( \triangle DCB \), and we can show another pair of angles equal using the above. But for the given table, we filled the reasons as per the steps.)

Answer:

The reasons filled in order for the statements are: Given, Given, Given, Triangle Sum Theorem, Substitution Property, Simplify, Subtraction Property, Substitution Property, Definition of Congruence, Reflexive Property, AA Similarity. (For the specific table cells, each statement's reason is filled as per the step-by-step above.)