Given: ∠A B C: R S is the perpendicular bisector of A B ; R T is the perpendicular bisector

step 1 Number 41, we are given the triangle or angle ABC. RS is the perpendicular bisector of line AB.

Given: ∠A B C: R S is the perpendicular bisector of A B ; R...

Key Concepts

Geometric Ratios

Geometric ratios compare the lengths of segments and are used to establish proportional relationships within figures. Proving such ratios often involves showing the equality of segments through congruence or symmetry, which then allows one to set up and solve proportional relationships. This key concept is central when a proof requires showing that one segment is a specific multiple of another.

Congruent Triangles

The concept of congruent triangles involves showing that two triangles are identical in terms of side lengths and angles. In geometric proofs, congruence criteria—such as the Hypotenuse-Leg (HL) theorem for right triangles, or Side-Angle-Side (SAS)—are used to deduce equalities between segments and angles, leading to the conclusion that ratios of segment lengths are equal or other required proportional relationships hold.

Perpendicular Bisector

A perpendicular bisector of a line segment is a line that divides the segment into two equal parts at a right angle. This concept is fundamental in geometry because any point on the perpendicular bisector is equidistant from the segment's endpoints, which is a critical property used to prove congruences and equalities in geometric constructions.

Midpoint

A midpoint is the point that divides a line segment into two segments of equal length. In many geometric proofs, establishing that a point is a midpoint helps in showing that certain segments are congruent or that triangles have corresponding equal sides, which is essential for further reasoning about the figure.

Transcript

00:01 Number 41, we are given the triangle or angle abc.

00:07 Rs is the perpendicular bisector of line ab.

00:10 That means that line rs is intersecting at a 90 degree angle and cutting it in half.

00:18 Rt is the perpendicular bisector of bc.

00:22 We need to prove that line a -r is congruent to line rc in this image.

00:30 First thing that we do in our two column proof is we start with our givens.

00:36 We are given that abc is an angle.

00:42 We are given that rs, the segment rs, is the perpendicular, which means it's intersecting at 90 degrees, bisector, cutting it in half, of line ab.

01:02 And we are given that rt is also the perpendicular 90 degrees bisector bc anytime we're given something it's usually good to state what that gives you our s being the perpendicular bisector well the definition of perpendicular means it's intersecting at a 90 degrees that means angle rsa and angle r s b equal 90 degrees that's the definition of b perpendicular.

01:47 And if rsa and r .b both equal 90 degrees, that means that rsa is congruent to angle r .s .b.

02:01 That is through the transitive property.

02:07 Transitive property say that if two things equal that same thing, they will equal each other.

02:15 Still talking about r .s.

02:18 And a .b.

02:19 We know that bisector means cut in half.

02:23 That means line sb, segment sb, is going to be equal to s .a.

02:32 They are cut in half.

02:34 That's the definition of bisect.

02:40 When you bisect something, you're cutting it in half.

02:43 So that was with dealing with the given from number two.

02:47 In number three of our two column proof, we're going to be saying the same things about the other triangle...