Get Ready: Module 2 — Justifying Mathematical Ideas & Arguments
Before we build the proofs, we pour the floor. This page rebuilds the few skills you need first — the everyday angle facts that every justification in Module 2 leans on. There is no rush and no wrong starting point. Read one card, try one problem, peek at the answer, and move on when you're ready.
Everyone starts somewhere. If geometry hasn't felt easy before, that says nothing about whether you can do it — only that no one has built the floor with you yet. We'll do that together, one solid plank at a time. Go at your own pace.
Foundations · 5 skills · Self-paced
Skills to Build First
Five short skills. Each one has a worked example you can follow, then a couple to try on your own.
Angle Types & Measures
What it is & why you need it An angle is measured in degrees. There are four names to know by their size: an acute angle is less than \(90^{\circ}\), a right angle is exactly \(90^{\circ}\) (the little square corner), an obtuse angle is between \(90^{\circ}\) and \(180^{\circ}\), and a straight angle is exactly \(180^{\circ}\) (a flat line). In Module 2 you'll justify steps by naming what kind of angle you have — so you need to recognize each on sight.
Worked example
Classify each angle: \(48^{\circ}\), \(90^{\circ}\), \(117^{\circ}\), and \(180^{\circ}\).
- \(48^{\circ}\) is less than \(90^{\circ}\) → acute.
- \(90^{\circ}\) is exactly \(90^{\circ}\) → right.
- \(117^{\circ}\) is between \(90^{\circ}\) and \(180^{\circ}\) → obtuse.
- \(180^{\circ}\) is exactly \(180^{\circ}\) → straight.
Try it
1. What type of angle is \(73^{\circ}\)?
Show answer
\(73^{\circ}\) is less than \(90^{\circ}\).
Answer: acute.
2. An angle measures \(134^{\circ}\). Is it acute, right, obtuse, or straight?
Show answer
\(134^{\circ}\) is greater than \(90^{\circ}\) but less than \(180^{\circ}\).
Answer: obtuse.
Angle Addition
What it is & why you need it When a bigger angle is split into smaller angles that sit side by side, the small angles add up to the big one. The two most useful versions: angles that form a straight line add to \(180^{\circ}\) (supplementary), and angles that form a square corner add to \(90^{\circ}\) (complementary). Almost every parallel-lines proof in Module 2 uses "these two angles add to \(180^{\circ}\)," so adding angles confidently is essential.
Worked example
Two angles sit side by side along a straight line. One is \(125^{\circ}\). Find the other.
- Angles on a straight line add to \(180^{\circ}\): \(\;125^{\circ} + x = 180^{\circ}\).
- Subtract \(125^{\circ}\) from both sides: \(\;x = 180^{\circ} - 125^{\circ}\).
- So \(x = 55^{\circ}\).
Try it
1. Two angles form a right angle (\(90^{\circ}\) total). One is \(32^{\circ}\). Find the other.
Show answer
They add to \(90^{\circ}\): \(\;32^{\circ} + x = 90^{\circ}\).
\(x = 90^{\circ} - 32^{\circ}\).
Answer: \(x = 58^{\circ}\).
2. A straight line is split into three angles: \(40^{\circ}\), \(70^{\circ}\), and \(x\). Find \(x\).
Show answer
All three add to \(180^{\circ}\): \(\;40^{\circ} + 70^{\circ} + x = 180^{\circ}\).
\(110^{\circ} + x = 180^{\circ}\), so \(x = 180^{\circ} - 110^{\circ}\).
Answer: \(x = 70^{\circ}\).
Solving an Equation for an Unknown Angle
What it is & why you need it Often an angle is written as an expression like \(2x + 10\) instead of a plain number. To find the angle, you first solve for \(x\) using the same "do the same thing to both sides" moves from algebra, then plug \(x\) back in. In Module 2, proofs frequently end with an algebra step like this — so getting one unknown out of an equation is a must.
Worked example
An angle of \(3x + 15\) degrees is supplementary to a \(60^{\circ}\) angle (they add to \(180^{\circ}\)). Find the angle.
- Set up the equation: \(\;(3x + 15) + 60 = 180\).
- Combine the numbers: \(\;3x + 75 = 180\).
- Subtract \(75\) from both sides: \(\;3x = 105\).
- Divide both sides by \(3\): \(\;x = 35\).
- Plug back in: the angle \(= 3(35) + 15 = 105 + 15 = 120^{\circ}\).
Try it
1. Solve for \(x\): \(\;2x + 20 = 90\).
Show answer
Subtract \(20\): \(\;2x = 70\).
Divide by \(2\): \(\;x = 35\).
Answer: \(x = 35\).
2. An angle of \(4x - 8\) degrees equals \(100^{\circ}\). Find \(x\), then state the angle.
Show answer
\(4x - 8 = 100\). Add \(8\): \(\;4x = 108\).
Divide by \(4\): \(\;x = 27\).
Check the angle: \(4(27) - 8 = 108 - 8 = 100^{\circ}\).
Answer: \(x = 27\), and the angle is \(100^{\circ}\).
Parallel vs. Perpendicular
What it is & why you need it Two lines are parallel if they run in the same direction and never cross — like the two rails of a train track (symbol: \(\parallel\)). Two lines are perpendicular if they cross to make a right angle, \(90^{\circ}\) (symbol: \(\perp\)). The whole point of Module 2's biggest theorems is what happens when a third line crosses two parallel lines — so you need to tell these two relationships apart instantly.
Worked example
Decide whether each describes parallel or perpendicular lines.
- The top and bottom edges of this screen never meet → they run the same direction → parallel (\(\parallel\)).
- The corner of a sheet of paper forms a \(90^{\circ}\) angle → the two edges are perpendicular (\(\perp\)).
- A line crosses another and the angle between them is \(90^{\circ}\) → perpendicular.
Try it
1. Two lines meet and form a \(90^{\circ}\) angle. Are they parallel or perpendicular? What symbol do you use?
Show answer
A \(90^{\circ}\) crossing means a right angle.
Answer: perpendicular, symbol \(\perp\).
2. Two lines stay the same distance apart forever and never touch. Parallel or perpendicular?
Show answer
Same distance apart, never crossing, is the definition of parallel.
Answer: parallel, symbol \(\parallel\).
Basic Triangle Facts (Angle Sum = \(180^{\circ}\))
What it is & why you need it The three inside angles of any triangle always add up to \(180^{\circ}\) — no matter how stretched or squished the triangle looks. If you know two of the angles, you can always find the third by subtracting from \(180^{\circ}\). Module 2 uses this fact constantly to justify steps and to find missing angles inside figures.
Worked example
A triangle has angles \(50^{\circ}\), \(60^{\circ}\), and \(x\). Find \(x\).
- The three angles add to \(180^{\circ}\): \(\;50^{\circ} + 60^{\circ} + x = 180^{\circ}\).
- Combine the known angles: \(\;110^{\circ} + x = 180^{\circ}\).
- Subtract \(110^{\circ}\) from both sides: \(\;x = 70^{\circ}\).
Try it
1. A triangle has angles \(90^{\circ}\) and \(35^{\circ}\). Find the third angle.
Show answer
\(90^{\circ} + 35^{\circ} + x = 180^{\circ}\), so \(125^{\circ} + x = 180^{\circ}\).
\(x = 180^{\circ} - 125^{\circ}\).
Answer: \(x = 55^{\circ}\).
2. In a triangle, one angle is \(x\), another is \(2x\), and the third is \(60^{\circ}\). Find \(x\).
Show answer
\(x + 2x + 60 = 180\), so \(3x + 60 = 180\).
Subtract \(60\): \(\;3x = 120\). Divide by \(3\): \(\;x = 40\).
Answer: \(x = 40^{\circ}\) (the angles are \(40^{\circ}\), \(80^{\circ}\), \(60^{\circ}\)).
Quick Readiness Check
Six short questions across all five skills. Try each in your head or on scratch paper, then check.
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What type of angle is \(95^{\circ}\)?
Show answer
Obtuse (it is between \(90^{\circ}\) and \(180^{\circ}\)).
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Two angles sit on a straight line. One is \(110^{\circ}\). What is the other?
Show answer
\(180^{\circ} - 110^{\circ}\).
\(70^{\circ}\).
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Solve for \(x\): \(\;5x - 10 = 90\).
Show answer
Add \(10\): \(\;5x = 100\). Divide by \(5\).
\(x = 20\).
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Two lines cross at a \(90^{\circ}\) angle. What is this relationship called?
Show answer
Perpendicular (\(\perp\)).
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A triangle has angles \(45^{\circ}\) and \(55^{\circ}\). Find the third angle.
Show answer
\(180^{\circ} - 45^{\circ} - 55^{\circ}\).
\(80^{\circ}\).
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An angle of \(2x + 30\) degrees is complementary to a \(40^{\circ}\) angle (they add to \(90^{\circ}\)). Find \(x\).
Show answer
\((2x + 30) + 40 = 90\), so \(2x + 70 = 90\).
Subtract \(70\): \(\;2x = 20\). Divide by \(2\).
\(x = 10\).
If these feel comfortable, you're ready for the module. If a couple felt shaky, that's completely fine — scroll back up, re-read that one skill card, and try the practice links below. Then come back. You've got this.
Practice More (Free)
Want more reps on any skill? These free sites let you drill as long as you like.
Angle Types & Measures
Angle Addition
Solving for an Unknown Angle
Parallel vs. Perpendicular
Triangle Angle Sum
When You're Ready →
The floor is poured. Step into the module whenever you feel steady.
Stuck or want a hand? That's what office hours are for.
Instructor: Dr. Goodluck Ijezie-Desbois, PharmD · Beta Academy
Reach out by appointment, at
gijezie-desbois@betaacademy.org,
or through ParentSquare.