TSI Geometry Workout Worksheet
TSI Assessment 2.0 Geometry & Measurement
TSI Geometry Workout
Name:
Date: Score:
1
Angles, Parallel Lines & Notation
Supp: \(\alpha + \beta = 180^\circ\) • Comp: \(\alpha + \beta = 90^\circ\) • Vert \(\angle s\): \(\alpha = \beta\)
Item 1. Lines \(l_1\) and \(l_2\) are parallel and cut by transversal \(t\). Two alternate interior angles are given by \(\angle 1 = (4x + 12)^\circ\) and \(\angle 2 = (7x - 24)^\circ\). Determine the value of \(x\) and find the measure of consecutive interior angle \(\angle 3\), which is supplementary to \(\angle 2\).
l₁ l₂ t ∠1 ∠2 ∠3
Show Work / Algebraic Steps
\(x =\)
\(m\angle 3 =\)
Item 2. In the figure, ray \(\vec{BD}\) bisects right angle \(\angle ABC\). Ray \(\vec{BE}\) lies in the interior of \(\angle DBC\) such that \(m\angle DBE = (3x - 1)^\circ\) and \(m\angle EBC = (2x + 6)^\circ\). Find the value of \(x\) and calculate the exact measure of \(\angle ABE\).
A B C D E
Show Work / Algebraic Steps
\(x =\)
\(m\angle ABE =\)
Item 3. Three straight lines intersect at point \(P\). The vertical angles and adjacent angles are labeled in terms of variables \(a\) and \(b\). If \(\angle 1 = (5a + 10)^\circ\), \(\angle 2 = (2b + 15)^\circ\), and their vertical counterparts satisfy \(\angle 3 = (3a + 40)^\circ\) and \(\angle 1 + \angle 2 + \angle 4 = 180^\circ\) where \(\angle 4 = 45^\circ\), solve for \(a\) and \(b\).
Show Work / Algebraic Steps
\(a =\)
\(b =\)
TSI Math Prep • Section 1: Angles & Lines Page 1 of 4
TSI Assessment 2.0 Properties, Perimeter & Area
Triangles & Polygons
Section Score: _____ / 3
2
Key Formulas
\(a^2+b^2=c^2\) • \(A_{\triangle}=\frac{1}{2}bh\) • \(\sum \angle_{int}=(n-2)\cdot 180^\circ\) • \(A_{trap}=\frac{a+b}{2}h\)
Item 4. A 25-foot maintenance ladder is propped against a vertical wall, with its base resting 7 feet away from the foundation of the wall. If the top of the ladder slips vertically downward by exactly 4 feet, how many feet will the base of the ladder slide outward away from the wall?
25 ft 7 ft 4 ft d = ?
Show Work / Geometric Modeling
New Base Distance:
Distance Slid Outward:
Item 5. Trapezoid \(ABCD\) has horizontal parallel bases \(\overline{AB}\) and \(\overline{DC}\). The base \(AB = 10\text{ cm}\) and non-parallel leg \(AD = 8\sqrt{2}\text{ cm}\). If \(\angle D = 45^\circ\) and \(\angle C = 30^\circ\), find the height \(h\) of the trapezoid, the length of leg \(\overline{BC}\), and the total area of the trapezoid.
AB = 10 8√2 45° 30° h
Show Work / Special Right Triangles Decomposition
Height \(h =\)
Leg \(BC =\)
Area \(=\)
Item 6. In a regular nonagon (9-sided polygon):
(a) Calculate the measure of each interior angle and each exterior angle.
(b) If each side of the nonagon measures \(6\text{ cm}\), what is its perimeter?
Show Work / Polygon Formulas
Interior \(\angle =\)
Exterior \(\angle =\)
Perimeter \(=\)
TSI Math Prep • Section 2: Triangles & Polygons Page 2 of 4
TSI Assessment 2.0 Circumference, Area & Volume
Circles & 3D Solids
Section Score: _____ / 3
3
Key Formulas
\(A = \pi r^2\) • \(C = 2\pi r\) • \(V_{cyl} = \pi r^2 h\) • \(V_{cone} = \frac{1}{3}\pi r^2 h\) • \(V_{sph} = \frac{4}{3}\pi r^3\)
Item 7. Circle \(O\) has radius \(r = 12\text{ cm}\) with central angle \(\angle AOB = 75^\circ\) intercepting minor arc \(\widehat{AB}\).
(a) Find the exact arc length of \(\widehat{AB}\) in terms of \(\pi\).
(b) Find the exact area of the shaded sector in terms of \(\pi\).
O A B 75° r=12
Show Work / Arc Length & Sector Area Formulas
Arc Length \(\widehat{AB} =\)
Sector Area \(=\)
Item 8. A cylindrical storage tank has radius \(6\text{ ft}\) and height \(14\text{ ft}\). The liquid from this full tank is poured into right circular conical containers with diameter \(6\text{ ft}\) and height \(8\text{ ft}\). How many complete conical containers can be filled to capacity?
r=6, h=14 d=6, h=8
Show Work / 3D Solid Volume Calculation
Cylinder Volume \(=\)
Full Containers \(=\)
Item 9. A solid wooden block is a rectangular prism with length \(10\text{ in}\), width \(6\text{ in}\), and height \(4\text{ in}\). A cylindrical hole of radius \(1.5\text{ in}\) is drilled through the center from top to bottom. Find the exact total surface area of the remaining composite solid in terms of \(\pi\).
Show Work / Composite Surface Area
Prism External Area:
Total Surface Area:
TSI Math Prep • Section 3: Circles & Solids Page 3 of 4
TSI Assessment 2.0 Distance, Midpoint & Slope
Coordinate Geometry
Section Score: _____ / 3
4
Key Formulas
\(d = \sqrt{(\Delta x)^2 + (\Delta y)^2}\) • \(M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\) • \(m = \frac{\Delta y}{\Delta x}\) • \(m_{\perp} = -\frac{1}{m}\)
Item 10. Segment \(\overline{AB}\) has endpoints at \(A(-5, 3)\) and \(B(3, -1)\).
(a) Calculate the exact length of \(\overline{AB}\) in radical form.
(b) Find coordinates of midpoint \(M\) of \(\overline{AB}\).
(c) Write the equation of the perpendicular bisector in slope-intercept form (\(y = mx + b\)).
Show Work / Distance, Midpoint & Perpendicular Slope
\(d =\)
\(M =\)
Eq:
Item 11. Quadrilateral \(PQRS\) has vertices \(P(-2, 1)\), \(Q(2, 4)\), \(R(5, 0)\), and \(S(1, -3)\).
(a) Compute slopes of all four sides to verify whether \(PQRS\) is a rectangle.
(b) Calculate side lengths \(\overline{PQ}\) and \(\overline{QR}\), then determine its area.
P Q R S
Show Work / Slope & Distance Verification
\(m_{PQ} =\)
\(m_{QR} =\)
Type:
Area \(=\)
Item 12. A circle in the \(xy\)-coordinate plane has diameter endpoints at \(C(2, -4)\) and \(D(-6, 2)\). Find the center of the circle, the exact radius \(r\), and write the standard equation of the circle in the form \((x - h)^2 + (y - k)^2 = r^2\).
Show Work / Circle Center, Radius & Standard Equation
Center \((h, k) =\)
Radius \(r =\)
Equation:
TSI Math Prep • Section 4: Coordinate Geometry Page 4 of 4
