TenMarks teaches you how to apply algebraic proofs in geometry.
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Learn about Algebraic Proofs in Geometry In this lesson, let's learn about algebraic proofs in geometry. We will learn about properties that justify statements. So before we get to it, let's talk about properties of congruence because in geometry when we have two different figures, if figure A is the same as figure A, this is called a congruence symbol. The reflexive property is congruence says figure A is always congruent with itself which makes sense; short name, reflexive property. Symmetric property says if figure A is congruent to figure B, if they are—numbers are equal, the figures are congruent. So, if figure A and figure B are congruent, then figure B is also congruent with figure A. That makes sense; if figure A and figure B are congruent, then both sides become true. The transitive property is something you got to remember. If figure A is congruent to figure B, if figure B is congruent to figure C, then A is also congruent to figure C. It makes sense; A = B, B = C, then A = C. But instead of equal, we've got congruence because while numbers are equal, figures are measurements of angles etcetera are congruent. So let's use these properties to determine which one would be use to justify these statements. A line segment AB is congruent with line segment CD. If AB is congruent with CD, then CD is also congruent with AB. Well that is the symmetric property of congruence. So this would be the symmetric property. If angle A is congruent to angle B, angle B is congruent to angle C, then A is congruent to C. well that would be the transitive property. Angle 1 is congruent to itself; well, that’s the reflexive property. Let's actually use these in some ways to solve for Y. We’re going to try and find Y. So the way we’re going to do this is you see this line segment JM, MK, and JK. Well, based on these, what do I know? Well, I know that this entire line segment or JK is equal to JM and MK. That’s the additive postulate. So, equals JM + MK; so what is JK? JK is 5Y – 4 = JM is Y + 3 + 2Y – 1. So before we start, let's simplify. We’re going to combine like terms. So Y and 2Y becomes 3Y and three and one becomes two, 3 – 1 becomes two. So 5Y – 4 = 3Y + 2. Now what I'm going to do is we are going to basically subtract—subtraction property, what we’re going to do is subtract 3Y from both sides. What am I left with? 5Y – 3Y is 2Y – 4 = this is 0 + 2. This is simplifying it. Then, use the additive property and I'm going to add four to both sides. So the left side will be 2Y = 2 + 4 = 6. This is again simplifying it. Then, what do we have to do? Well, let's use the divisive or division property and divide by two on both sides. 2Y ÷ 2 is Y, 6 ÷ 2 is 3. This is simplified. So, we've solved for Y. Notice all we did was remembered that the segment addition postulate tells us that two segments are given to us, the length of the two segments. Then the length of the segment that makes up both the segments equals to the sum of both. We used that and then simplified it. Solve the equation using the properties we know of addition, subtraction, division and multiplication etcetera.