Geometry

Big idea: every figure can be located, moved, and matched — transformations are the language we use to say two shapes are the same.

Topics

• Geometry ReasoningG.4A,C,D · G.9B
• Using a Rectangular Coordinate SystemG.2B,C · G.3C · G.4B · G.5A–C · G.9B · G.10B · G.11A,B
• Sequences of Rigid MotionsG.3A–C · G.5B · G.6A,C
• Congruence Through TransformationsG.2B · G.3D · G.4C · G.5A · G.6B,C

You'll be able to…

• Use definitions, postulates, and conjectures to reason about points, lines, and planes.
• Find distance, midpoint, and slope between coordinate points and use them to classify figures.
• Perform and compose translations, rotations, and reflections as algebraic rules.
• Prove two figures are congruent by mapping one onto the other with a sequence of rigid motions.

Worked example — distance The distance between \(A(1,2)\) and \(B(4,6)\) is \[ AB=\sqrt{(4-1)^2+(6-2)^2}=\sqrt{9+16}=\sqrt{25}=5. \]

Worked example — composing rigid motions Reflect over the \(x\)-axis, then translate right 3: \((x,y)\to(x,-y)\to(x+3,\,-y)\).

Big idea: "it looks true" is not enough — a geometric claim earns belief only when every step is justified by a definition, postulate, or theorem.

Topics

• Composing & Decomposing ShapesG.4A,B · G.5A–D · G.6E · G.9B
• Justifying Line & Angle RelationshipsG.3B · G.4A–C · G.5A–D · G.6A,B,D · G.9B · G.12A
• Using Congruence TheoremsG.4B · G.5A–C · G.6B,E · G.12A

You'll be able to…

• Compose and decompose polygons to derive area and angle relationships.
• Justify relationships among angles formed by parallel lines and a transversal.
• Write two-column, paragraph, and flow proofs that defend a claim.
• Apply SSS, SAS, ASA, AAS, and HL to prove triangles congruent.
• Use CPCTC to prove that corresponding parts of congruent triangles are equal.

Worked example — angle sum The interior angles of any triangle satisfy \[ \angle A+\angle B+\angle C=180^{\circ}. \] So if \(\angle A=52^{\circ}\) and \(\angle B=71^{\circ}\), then \(\angle C=180^{\circ}-52^{\circ}-71^{\circ}=57^{\circ}.\)

Worked example — vertical angles When two lines cross, vertical angles are congruent: if one angle is \((3x+10)^\circ\) and its vertical partner is \((5x-20)^\circ\), then \(3x+10=5x-20\Rightarrow x=15.\)

Big idea: scaling a figure preserves its shape but not its size — and those fixed ratios power similarity and right-triangle trigonometry.

Topics

• SimilarityG.2A,B · G.3A–C · G.4C · G.5B,C · G.6A,D · G.7A,B · G.8A,B
• TrigonometryG.7B · G.8A · G.9A,B

You'll be able to…

• Use dilations to produce similar figures and identify the scale factor.
• Apply AA, SSS, and SAS similarity criteria to prove triangles similar.
• Set up and solve proportions from corresponding sides.
• Use \(\sin\theta,\ \cos\theta,\) and \(\tan\theta\) to find missing sides and angles in right triangles.

Worked example — trig ratio In a right triangle with the angle \(\theta\), opposite \(=3\), hypotenuse \(=5\): \[ \sin\theta=\frac{\text{opp}}{\text{hyp}}=\frac{3}{5}=0.6,\qquad \theta=\sin^{-1}(0.6)\approx 36.9^{\circ}. \]

Worked example — similar triangles If \(\triangle ABC\sim\triangle DEF\) with scale factor \(k=\tfrac{3}{2}\) and \(DE=8\), then \(AB=\tfrac{3}{2}\cdot 8=12.\)

Big idea: circles and solids aren't just pictures — they're equations, and equations let us measure what we can't easily draw.

Topics

• CirclesG.2B · G.3C · G.5A–C · G.11B · G.12A–E
• Building Three-Dimensional ShapesG.10A,B · G.11C,D

You'll be able to…

• Write and graph the equation of a circle, \((x-h)^2+(y-k)^2=r^2\).
• Relate central angles, inscribed angles, arcs, and arc length.
• Compute surface area and volume of prisms, cylinders, cones, pyramids, and spheres.
• Identify cross-sections and solids of revolution from 2-D figures.

Worked example — equation of a circle A circle centered at \((2,-3)\) with radius \(4\) has equation \[ (x-2)^2+(y+3)^2=16. \]

Worked example — prism volume & surface area For a box with \(\ell=4,\ w=3,\ h=2\): \[ V=\ell w h=4\cdot 3\cdot 2=24,\qquad SA=2(\ell w+\ell h+wh)=2(12+8+6)=52. \] (Try this live in the 3-D Solid tool above.)

Big idea: probability turns uncertainty into a number you can reason with — and geometry's precision carries straight into how we measure chance.

Topics

• Independence & Conditional ProbabilityG.13A,C,D,E
• Computing ProbabilitiesG.13A–E

You'll be able to…

• Describe sample spaces and events using set language and Venn diagrams.
• Determine whether two events are independent.
• Compute conditional probabilities, \(P(A\mid B)=\dfrac{P(A\cap B)}{P(B)}\).
• Apply the addition and multiplication rules to real decisions.

Worked example — conditional probability If \(P(A\cap B)=0.12\) and \(P(B)=0.40\), then \[ P(A\mid B)=\frac{0.12}{0.40}=0.30. \]

Worked example — independent events For independent events, \(P(A\cap B)=P(A)\,P(B)\). Two fair coin flips both heads: \(P=\tfrac{1}{2}\cdot\tfrac{1}{2}=\tfrac{1}{4}.\)