Math  /  Geometry

QuestionWE DO: Write a proof. Given BB is the midpoint of line AD;ABCA D ; \angle A B C and DBC\angle D B C are right angles, Prove ABC=DBC\triangle A B C=D B C

Studdy Solution

STEP 1

What is this asking? We're given a line segment AD with a midpoint B, and two right angles formed at B.
We need to prove that the two triangles ABC and DBC are congruent! Watch out! Don't assume anything beyond what's given!
We need to stick to the facts to build our proof.

STEP 2

1. State the givens
2. Prove congruence

STEP 3

Alright, let's **lay down the foundation**!
We're given that B is the **midpoint** of line segment AD.
What does that even mean?
It means that AB and BD are **equal**!
So, we can write AB=BDAB = BD.
This is our **first key** to unlocking this proof!

STEP 4

Next up, we're told that ABC\angle ABC and DBC\angle DBC are **right angles**.
That's awesome!
This tells us two things.
First, ABC=DBC\angle ABC = \angle DBC, because all right angles are equal.
Second, they both measure ABC=DBC=90\angle ABC = \angle DBC = 90^\circ.
This is our **second key**!

STEP 5

Now, let's talk about the **triangles**!
Notice that both triangles ABC and DBC **share a side**, BC.
That means BC is equal to itself, or BC=BCBC = BC.
This might seem obvious, but it's our **third key**!
It's called the **reflexive property**.

STEP 6

Let's **put it all together**!
We have AB=BDAB = BD (because B is the midpoint), ABC=DBC\angle ABC = \angle DBC (because they're both right angles), and BC=BCBC = BC (by the reflexive property).
Look closely!
We have two sides and the angle between them that are equal in both triangles.
This is the **Side-Angle-Side (SAS) postulate**!

STEP 7

Therefore, by the **SAS postulate**, we can confidently say that triangle ABC is congruent to triangle DBC.
We write this as ABCDBC\triangle ABC \cong \triangle DBC.
Boom!

STEP 8

We have proven that ABCDBC\triangle ABC \cong \triangle DBC using the SAS postulate, based on the given information.

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