1.1-The sum of lengths of all sections of a segment is smaller than the length of the segment; the sum of th emeasures of two smaller angles that form a larger angle is the measure of the larger angle.

1.2* - A construction is a geometric figure produced using only a straightedge and a compass. Constructions are useful tools in geometry.

1.3 - The midpoint formula is used to find the midpoint of a segment on the coordinate plane and the Distance Formula is used to find the length. The Midpoint Formula can be adapted to partition a segment into two segments with any ratio of lengths.

1.4 - Inductive reasoning can be used to identify patterns, provide evidence for or disprove conjectures, and make predictions.

1.5 - In mathematics, conditional statements are expressed as if-then statements consisting of a hypothesis and a conclusion. Conditional statements can be evaluated using truth tables.

1.6 - Deductive reasonsing is the process of using given statements or facts to logically reach a valid conclusion 

1.7 - Justify statements of a  proof with definitions, postulates, theorems, and properties

1.8* - In indirect reasoning, you assume that the disired conclusion is fale, which leads to a contradiction, allowing you to conclude that what you want to prove must be true.

2.1 -When parallel lines are cut by a transversal, the special angle pairs that are formed are congruent, supplementary, or both.

2.2 - Pairs of congruent or supplementary angles formed when two lines are cut by a transversal can be used to prove parallel lines.

2.3 - The sum of the interior angles of a triangle is 180 degrees, and the measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles.

2.4 - Two parallel lines have equal slopes. The product of the sloes of perpendicular lines is -1.

What properties are specific to parallel lines and perpendicular lines?

3.1 - Reflections are rigid motions across a line of reflection.

3.2 - A translation is a rigid motion that moves all points of the preimage the same distance in the same direction. A translation is the composition of two reflections.

3.3 - Rotation is a rigid motion described by its center of rotation and angle of rotation. Any rotation can be described by two reflections whose lines of reflection meet at the center of rtation at half the angle of rotation.

3.4 - Any composition of rigid motions can be represented by a combination of at least two of the following: a translation, reflection, rotation, or glide reflection.

3.5 - A figure that can be mapped onto itself using rigid motions is symmetric. Rotational symmetry uses rotation to mapa figure onto itself, and reflectional symmetry uses reflection to mpa a figure onto itself.

What are the properties of the four types of rigid motion?

4.1 - Figures that have the same size and shape are congruent. If a rigid motion or composition of rigid motions can map one figure onto another, then the figures are congruent. 

4.2 - An isosceles triangle has congruent base angles and legs. The angle bisector of the vertex angle of an isosceles triangle is also the perpendicular bisector of the base. An equilateral triangle is also equiangular. 

4.3  - Prove triangle congruence by SSS & SAS criteria and use triangle congruence to solve problems. Understand that corresponding parts of congruent triangles are congruent and use CPCTC.

4.4 - Prove triangle congruence by ASA & AAS criteria and use triangle congruence to solve problems.Understand that corresponding parts of congruent triangles are congruent and use CPCTC.

4.5  - Prove the Hypotenuse-Leg Theorem. Use congruence criteria for triangles to solve problems and to prove relationships in geometric figures. 

4.6  - Apply congruence criteria to increasingly intricate problems involving overlapping triangles and multiple triangles.  Prove triangles are congruent by identifying corresponding parts and applying the correct theorems. 

Why is the triangle a significant plane figure?

How are the properties and characteristics of triangles used in applications?

5.1 - The perpendicular bisector of a segment contains all the points that are equidistant from the endpoints of the segment, and an angle bisector contains all the points that are equidistant from the sides of the angle.

5.2 - The perpendicular bisectors of the sides of a triangle are concurrent at its circumcenter. The angle bisectors of a triangle are concurrent at its incenter. 

5.3 - The medians of a triangle are concurrent at its centroid. The lines containing the altitudes of a triangle are concurrent at its orthocenter. 

5.4 -The lengths of the sides of a triangle are related to the measures of the angles in the triangle. The sum of the lengths of two sides of a triangle is greater than the length of the third side. 

5.5 - When two triangles have two pairs of congruent sides, the longer third side is opposite the larger included angle, and the shorter third side is opposite the smaller included angle. 

Triangles have important characteristics and properties.

How are the properties and characteristics of triangles used in applications?

What properties are common to all triangles?
 

6.1 - The sum of the exterior angles of a polygon is 360°, regardless of the number of sides. The sum of the interior angles of a polygon is 180° • (n – 2), where n is the number of sides. 

6.2 - Diagonals of a kite are perpendicular, and one diagonal bisects the other. In isosceles trapezoids, diagonals are congruent. The length of the midsegment of a trapezoid is half the sum of the lengths of the bases.

6.3 - In a parallelogram, consecutive angles are supplementary, opposite angles are congruent, opposite sides are congruent, and the diagonals bisect each other. 

6.4 - A quadrilateral with two pairs of congruent opposite sides, or one pair of congruent parallel sides, or diagonals bisecting each other is a parallelogram. A quadrilateral with an angle supplementary to both of its consecutive angles, or two pairs of opposite congruent angles is a parallelogram. 

6.5 - The diagonals of a rhombus are perpendicular, bisect each other, and bisect opposite angles. They form four congruent triangles. In a rectangle, the diagonals are congruent. Squares have properties of rhombuses and rectangles. 

6.6 - A parallelogram with perpendicular diagonals or diagonals that bisect angles is a rhombus. A parallelogram with congruent diagonals is a rectangle. A parallelogram with perpendicular congruent diagonals or with congruent diagonals and a diagonal that bisects angles is a square. 

What are the unique properties and characteristics associated with individual geometric figures?

What are the properties and theorems that connect two or more geometric figures (e.g. congruence, similarity, etc)?

How can we use the properties and characteristics of figures to do real-world problems?

7-1- A dilation is a transformation that preserves angle measure but not length. The dilation of a figure is determined by the scale factor and center of dilation. Every distance from the center of dilation and every side length in a preimage are multiplied by the scale factor to find the corresponding distance and side length in the image.

7-2- A similarity transformation is a dilation combined with one or more rigid motions. In order for two figures to be similar, there must be a similarity transformation that maps one figure to the other. All circles are similar. 

7-3- Two triangles are similar if a composition of rigid motions and dilation will map one triangle onto the other. Two pairs of congruent angles, or three pairs of sides with lengths that are in the same proportion, or two pairs of sides having congruent included angles with lengths that are in the same proportion, are sufficient to show that two triangles are similar. 

7-4- The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the two segments into which the altitude divides the hypotenuse. The length of a leg of a right triangle is the geometric mean of the length of the hypotenuse and the segment of the hypotenuse adjacent to the leg.

7-5- A segment parallel to one side of a triangle divides the triangle into two similar triangles. If that segment connects the midpoints of two sides, the smaller triangle is in proportion 1 : 2 with the larger triangle. A segment that bisects an angle of a triangle divides the opposite side of the triangle into segments that are proportional to the adjacent sides.

Why is it essential to understand the interconnectedness of geometric properties, similarity, ratios, proportions, and transformations to effectively model and solve problems across mathematics and in real-world scenarios?

The Pythagorean Theorem can be understood through the relationships among the similar triangles formed by the altitude to the hypotenuse. The length of the hypotenuse of a 45-45-90 triangle is rt(2) times the leg length. The length of the hypotenuse of a 30-60-90 right triangle is twice the length of the shorter leg, and the length of the longer leg is rt(3) times the length of the shorter leg.

8.2 - For any two right triangles with a given acute angle, the ratios of any two corresponding side lengths are equal. The ratio of the opposite side to the ypotenuse is the sine ratio, the ratio of the ajdacent side to the hypotenuse is the cosine ratio, and the ratio of the opposite side to the adjacent side is the tangent ratio.

8.5 - The ratios of the corresponding sides of right triangles are constant for right triangles with given base angles and are related to the base angles. These relationships can be used to solve problems for which side lengths, angle measures, or areas of triangles are desired.

How are the Pythagorean Theorem and trigonometry useful?

9.1 - Algebra is used to determine properties of geometric figures drawn on the coordinate plane. Slopes can be used to determine whether segments are parallel or perpendicular, the distance formula can be used to find length of segments, and the Midpoint Formula can be used to bisect segments.

9.2 - Proofs using coordinate geometry require planning by determining the properties to be shown with algebra, drawing a labeled diagram on the coordinate plane and calculating the values needed to show the desired relationships.

9.3 - The equation of a circle in the coordinate plane is given by (x-h)^2-(y-k)^2=r^2, where (h,k) is the center of the circle and r is the radius.

How can geometric relationships be proven by applying algebraic properties to geometric figures represented in the coordinate plane?

10.1 - Arcs are classified as minor arcs or major arcs depending on whether they are small or larger than a semicircle. The length of an arc is a part of the circumference that is proportional to the corresponding central angle. The area o f a sector of a circle is part of the area of the circle that is proportional to the central angle. The area of a segment of a circle is the area of the ocrrespnding sector minus the area of the corresponding triangle.

10.2 - A line that is tangnet to a circle intersects the circle at exactly one point and is perpendicular to the radius to that point. If two segments are tangent ot the same circle and have a common enpoint exterior to the cricle, the semgents are congruent.

10.3 - In a circle or congruent circles, two chords are congruent if the corresponding central angles are congruent or if the chords intercept congruent arcs. A diameter that bisects a chors is perpendicular to the chord.

10.4 - In a circle, the measure of an inscribed angle is one-half of the measure of its intercepted arc. In a circle, the measure of an angle formed by a chord and tangent ot the circle is one-half of the measure of its intercepted arc.

10.5 - When two secants intersect inside a circle, the measure of the angle formed is half the sum of the intercepted arcs. When secants or tangents intersect outside a cricle, the measure of the angle formed is half the difference of the intercepted arcs. When secants or tangents intersect inside or outside a circle, the products of the distances from the point of intersection to the points on a circle is the same for both lines.

How can we use the properties of arcs, tangents, chords, inscribed angles, and secants to analyze and solve problems involving circles?