Question

unit 3 lesson 3 foundations of euclidean geometry warm up 1. if ab = 12 and bc = 7 and b lies between a and c, find ac. 2. if m is the mid - point of ab, am = 3x + 1 and mb = 5x - 7. find x. define collinear points: define complementary angles: a is an idea that has to be shown to be true through a chain of reasoning, often called a proof. a is an idea that is assumed to be true that is something that is self - evident.

Explanation:

Step1: Use segment - addition postulate for first problem

Since point B lies between A and C, then \(AC = AB+BC\). Given \(AB = 12\) and \(BC = 7\), we substitute the values into the formula.
\(AC=12 + 7\)

Step2: Calculate the value of AC

\(AC=19\)

Step3: Use mid - point property for second problem

If M is the mid - point of AB, then \(AM = MB\). Given \(AM = 3x + 1\) and \(MB=5x - 7\), we set up the equation \(3x + 1=5x - 7\).

Step4: Solve the equation for x

First, subtract \(3x\) from both sides: \(1 = 5x-3x - 7\), which simplifies to \(1 = 2x-7\). Then add 7 to both sides: \(1 + 7=2x\), so \(8 = 2x\). Divide both sides by 2: \(x = 4\)

Step5: Define collinear points

Collinear points are points that lie on the same straight line.

Step6: Define complementary angles

Complementary angles are two angles whose sum is \(90^{\circ}\).

Step7: Define the terms for the last two blanks

A statement that is assumed to be true without proof is called an axiom or a postulate. A statement that has been proven to be true through a chain of reasoning is called a theorem.

Answer:

1. \(AC = 19\)
2. \(x = 4\)
3. Collinear points are points on the same straight line.
4. Complementary angles are two angles whose sum is \(90^{\circ}\).
5. Axiom/Postulate
6. Theorem