Answer:
To demonstrate a geometry problem we always have to mention the reason why we are doing each step. So, we have to now a few things:
- When we begin to demonstrate, we always start from the hypothesis, all relations that are true are justified as "by hypothesis", because they are being taken from the problem.
Also, there're some definitions which are important to understand the demonstration below.
Definition of midpoint: The midpoint is define as the middle between to extreme points, dividing the formed segment in two equal parts, for example, If C is mid point of the segment AB, then AC and CD are equal, because by definition, the midpoint divides equally to half.
Vertical angle theorem: This definition states that both angles that don't have any side in common, but just the vertex, they are congruent. Also, we could name them as opposite by vertex angles.
Reflexive property: This definition states that a segment is always congruent to itself, so segment AB is congruent with BA or AB. It's like saying that 2 is equal to 2.
SAS congruency postulate: this is one of the postulates to demonstrate congruence between triangle. Specifically, this postulate states that two triangles are congruent if the have two congruent corresponding sides and one congruent corresponding angle in between these sides.
Angle bisector: an angle bisector is a line that divides angles in equal parts. For example, is AB is bisector of angle C=100, then AB divides C in two equal parts of 50.
AAS congruence postulate: this is another postulate of congruency between triangles, this states that, if two triangles as two corresponding angles congruent, and one corresponding side congruent, then those triangles are congruent.
SSS congruence postulate: this is another congruence postulate, which states that if two triangles have all three sides congruent, then they are completely congruent.
Basically, to demonstrate a congruence, we just have to look what postulate is better to achieve success.
Statement Reason
1. L is midpoint of KN and MP 1. By hypothesis.
2. KL ≅ NL 2. Midpoint's defintion.
3. ∠KLM ≅ ∠NLP 3. Vertical Angle Theorem.
4. ML ≅ PL 4. Midpoint's defintion.
5. ΔMKL ≅ ΔPNL 5. SAS postulate of congruence.
Statement Reason
1. ∠BAD ≅ ∠BCD 1. By hypothesis.
2. BD bisects ∠ABC 2. By hypothesis.
3. ∠ABD ≅ ∠CBD 3. Definition of angle bisector
4. BD ≅ BD 4. Reflexive Property
5. ΔABD ≅ ΔCBD 5. AAS postulate of congruence.
Statement Reason
1. S is midpoint of QU 1. By hypothesis.
2. QS ≅ US 2. Midpoint's definition.
3. QR ≅ ST 3. By hypothesis.
4. RS ≅ TU 4. By hypothesis.
5. ΔQRS ≅ ΔSTU 5. SSS postulate of congruence.