<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
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</head>
<body>

  
  <div class="nav-bar">
    <button class="tab-btn active" data-step="0">Overview</button>
    <button class="tab-btn" data-step="1">AB ∥ CD</button>
    <button class="tab-btn" data-step="2">AD ∥ BC</button>
    <button class="tab-btn" data-step="3">Shared AC</button>
    <button class="tab-btn" data-step="4">ΔABC ≅ ΔCDA</button>
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        <text x="358" y="76">C</text>

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        <text x="122" y="76">D</text>
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  <div id="statusBadge" class="badge-card">
    Parallelogram ABCD with diagonal AC
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</body>
</html>

Properties and Proofs of Parallelograms

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<body>

  <div class="conclusion-tag" id="statusLabel">△ABE ≅ △CDE (ASA)</div>

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      <text x="396" y="268" class="label">C</text>
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      <text x="250" y="174" class="label">E</text>
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  </div>

  <div class="btn-group">
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    <button class="step-btn" onclick="setMode(1)">2. Diagonals Bisect</button>
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</html>

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<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
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      <button class="btn active" id="btnSlanted" onclick="setAngle(25)">Parallelogram (25°)</button>
    </div>
  </div>

  <script>
    const slider = document.getElementById('tiltSlider');
    const btnRect = document.getElementById('btnRect');
    const btnSlanted = document.getElementById('btnSlanted');

    const cy = 150;
    const width = 230;
    const height = 140;

    function renderChevron(x, y, angleRad, size = 6) {
      const cos = Math.cos(angleRad);
      const sin = Math.sin(angleRad);
      const perpCos = -sin;
      const perpSin = cos;

      const p1x = x - size * cos + size * 0.9 * perpCos;
      const p1y = y - size * sin + size * 0.9 * perpSin;
      const p2x = x;
      const p2y = y;
      const p3x = x - size * cos - size * 0.9 * perpCos;
      const p3y = y - size * sin - size * 0.9 * perpSin;

      return `<path d="M ${p1x} ${p1y} L ${p2x} ${p2y} L ${p3x} ${p3y}" />`;
    }

    function updateDiagram(deg) {
      const rad = (deg * Math.PI) / 180;
      const shift = Math.tan(rad) * (height / 2);

      // Points: A top-left, B top-right, C bottom-right, D bottom-left
      const topY = cy - height / 2;
      const botY = cy + height / 2;

      const Ax = 250 - width / 2 + shift;
      const Ay = topY;

      const Bx = 250 + width / 2 + shift;
      const By = topY;

      const Cx = 250 + width / 2 - shift;
      const Cy = botY;

      const Dx = 250 - width / 2 - shift;
      const Dy = botY;

      // Update Polygon
      document.getElementById('polyFill').setAttribute('points', `${Ax},${Ay} ${Bx},${By} ${Cx},${Cy} ${Dx},${Dy}`);

      // Update Edges
      const setLine = (id, x1, y1, x2, y2) => {
        const el = document.getElementById(id);
        el.setAttribute('x1', x1);
        el.setAttribute('y1', y1);
        el.setAttribute('x2', x2);
        el.setAttribute('y2', y2);
      };
      setLine('edgeAB', Ax, Ay, Bx, By);
      setLine('edgeBC', Bx, By, Cx, Cy);
      setLine('edgeCD', Dx, Dy, Cx, Cy);
      setLine('edgeDA', Dx, Dy, Ax, Ay);

      // Single arrows for AB (pointing right) and CD (pointing right from D to C)
      const midABx = (Ax + Bx) / 2;
      const midABy = (Ay + By) / 2;
      document.getElementById('arrowAB').innerHTML = renderChevron(midABx, midABy, 0, 7);

      const midCDx = (Dx + Cx) / 2;
      const midCDy = (Dy + Cy) / 2;
      document.getElementById('arrowCD').innerHTML = renderChevron(midCDx, midCDy, 0, 7);

      // Slanted angle for AD and BC
      const slantAngle = Math.atan2(Ay - Dy, Ax - Dx);

      // Double arrows for DA (pointing along D->A)
      const midDAx = (Dx + Ax) / 2;
      const midDAy = (Dy + Ay) / 2;
      const cosS = Math.cos(slantAngle);
      const sinS = Math.sin(slantAngle);
      const spacing = 7;
      document.getElementById('arrowDA').innerHTML =
        renderChevron(midDAx - (spacing/2) * cosS, midDAy - (spacing/2) * sinS, slantAngle, 6) +
        renderChevron(midDAx + (spacing/2) * cosS, midDAy + (spacing/2) * sinS, slantAngle, 6);

      // Double arrows for BC (pointing along C->B)
      const midBCx = (Cx + Bx) / 2;
      const midBCy = (Cy + By) / 2;
      document.getElementById('arrowBC').innerHTML =
        renderChevron(midBCx - (spacing/2) * cosS, midBCy - (spacing/2) * sinS, slantAngle, 6) +
        renderChevron(midBCx + (spacing/2) * cosS, midBCy + (spacing/2) * sinS, slantAngle, 6);

      // Vertices
      const setCircle = (id, cx, cy) => {
        const el = document.getElementById(id);
        el.setAttribute('cx', cx);
        el.setAttribute('cy', cy);
      };
      setCircle('ptA', Ax, Ay);
      setCircle('ptB', Bx, By);
      setCircle('ptC', Cx, Cy);
      setCircle('ptD', Dx, Dy);

      // Labels offset outwards
      document.getElementById('lblA').setAttribute('x', Ax - 14);
      document.getElementById('lblA').setAttribute('y', Ay - 14);

      document.getElementById('lblB').setAttribute('x', Bx + 14);
      document.getElementById('lblB').setAttribute('y', By - 14);

      document.getElementById('lblC').setAttribute('x', Cx + 14);
      document.getElementById('lblC').setAttribute('y', Cy + 14);

      document.getElementById('lblD').setAttribute('x', Dx - 14);
      document.getElementById('lblD').setAttribute('y', Dy + 14);

      // Button state highlights
      if (Math.abs(deg) < 1) {
        btnRect.classList.add('active');
        btnSlanted.classList.remove('active');
      } else if (Math.abs(deg - 25) < 1) {
        btnRect.classList.remove('active');
        btnSlanted.classList.add('active');
      } else {
        btnRect.classList.remove('active');
        btnSlanted.classList.remove('active');
      }
    }

    function setAngle(deg) {
      slider.value = deg;
      updateDiagram(deg);
    }

    slider.addEventListener('input', (e) => {
      updateDiagram(parseFloat(e.target.value));
    });

    // Initial render
    updateDiagram(25);
  </script>
</body>
</html>

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      user-select: none;
      -webkit-user-select: none;
    }
    body {
      width: 100%;
      height: 100%;
      background: #ffffff;
      font-family: system-ui, -apple-system, sans-serif;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      padding: 12px 16px;
      color: #1e2945;
    }
    .header-badge {
      font-size: 15px;
      font-weight: 700;
      color: #0284c7;
      background: #e0f2fe;
      padding: 4px 14px;
      border-radius: 999px;
      letter-spacing: 0.2px;
    }
    .svg-wrap {
      width: 100%;
      max-width: 460px;
      flex: 1;
      display: flex;
      align-items: center;
      justify-content: center;
    }
    svg {
      width: 100%;
      height: auto;
      overflow: visible;
    }
    .control-row {
      display: flex;
      align-items: center;
      gap: 12px;
      width: 100%;
      max-width: 380px;
      padding: 4px 8px;
    }
    .control-label {
      font-size: 13px;
      font-weight: 600;
      color: #64748b;
      white-space: nowrap;
    }
    input[type=range] {
      flex: 1;
      accent-color: #0284c7;
      cursor: pointer;
      height: 6px;
    }
  </style>
</head>
<body>

  <div class="header-badge" id="angleEq">θ = θ</div>

  <div class="svg-wrap">
    <svg id="diagram" viewBox="0 0 440 250" aria-label="Alternate interior angles in Z-shape">
      <defs>
        
        <marker id="arrow" viewBox="0 0 10 10" refX="5" refY="5" markerWidth="6" markerHeight="6" orient="auto-start-reverse">
          <path d="M 1 2 L 6 5 L 1 8" fill="none" stroke="#64748b" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" />
        </marker>
        <marker id="arrowBlue" viewBox="0 0 10 10" refX="5" refY="5" markerWidth="6" markerHeight="6" orient="auto-start-reverse">
          <path d="M 1 2 L 6 5 L 1 8" fill="none" stroke="#0284c7" stroke-width="2.5" stroke-linecap="round" stroke-linejoin="round" />
        </marker>
      </defs>

      
      <line x1="30" y1="65" x2="410" y2="65" stroke="#94a3b8" stroke-width="2.5" stroke-linecap="round" />
      <line x1="30" y1="185" x2="410" y2="185" stroke="#94a3b8" stroke-width="2.5" stroke-linecap="round" />

      
      <path d="M 85 60 L 93 65 L 85 70 M 95 60 L 103 65 L 95 70" fill="none" stroke="#64748b" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" />
      <path d="M 335 180 L 343 185 L 335 190 M 345 180 L 353 185 L 345 190" fill="none" stroke="#64748b" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" />

      
      <line id="transExtTop" stroke="#cbd5e1" stroke-width="2.5" stroke-dasharray="4 4" stroke-linecap="round" />
      <line id="transExtBot" stroke="#cbd5e1" stroke-width="2.5" stroke-dasharray="4 4" stroke-linecap="round" />

      
      <path id="zPath" fill="none" stroke="#0284c7" stroke-width="4" stroke-linecap="round" stroke-linejoin="round" />

      
      <path id="topArcFill" fill="#0284c7" fill-opacity="0.15" />
      <path id="topArc" fill="none" stroke="#0284c7" stroke-width="3" stroke-linecap="round" />
      <text id="topLabel" font-size="18" font-weight="700" fill="#0284c7" text-anchor="middle" dominant-baseline="central">θ</text>

      
      <path id="botArcFill" fill="#0284c7" fill-opacity="0.15" />
      <path id="botArc" fill="none" stroke="#0284c7" stroke-width="3" stroke-linecap="round" />
      <text id="botLabel" font-size="18" font-weight="700" fill="#0284c7" text-anchor="middle" dominant-baseline="central">θ</text>

      
      <circle id="pTop" r="4.5" fill="#0284c7" />
      <circle id="pBot" r="4.5" fill="#0284c7" />
    </svg>
  </div>

  <div class="control-row">
    <span class="control-label">Tilt</span>
    <input type="range" id="angleSlider" min="30" max="68" value="50" aria-label="Transversal angle slider">
  </div>

  <script>
    const yTop = 65;
    const yBot = 185;
    const cx = 220;
    const cy = 125;
    const arcR = 36;

    const slider = document.getElementById('angleSlider');
    const zPath = document.getElementById('zPath');
    const transExtTop = document.getElementById('transExtTop');
    const transExtBot = document.getElementById('transExtBot');
    const topArc = document.getElementById('topArc');
    const topArcFill = document.getElementById('topArcFill');
    const botArc = document.getElementById('botArc');
    const botArcFill = document.getElementById('botArcFill');
    const topLabel = document.getElementById('topLabel');
    const botLabel = document.getElementById('botLabel');
    const pTop = document.getElementById('pTop');
    const pBot = document.getElementById('pBot');
    const angleEq = document.getElementById('angleEq');

    function update(deg) {
      const rad = (deg * Math.PI) / 180;
      const dy = (yBot - yTop) / 2;
      const dx = dy / Math.tan(rad);

      const xTop = cx + dx;
      const xBot = cx - dx;

      // Z endpoints
      const zLeftTop = 70;
      const zRightBot = 370;

      // Draw Z shape
      zPath.setAttribute('d', `M ${zLeftTop} ${yTop} L ${xTop} ${yTop} L ${xBot} ${yBot} L ${zRightBot} ${yBot}`);

      // Transversal extensions
      const extLen = 28;
      transExtTop.setAttribute('x1', xTop);
      transExtTop.setAttribute('y1', yTop);
      transExtTop.setAttribute('x2', xTop + Math.cos(rad) * extLen);
      transExtTop.setAttribute('y2', yTop - Math.sin(rad) * extLen);

      transExtBot.setAttribute('x1', xBot);
      transExtBot.setAttribute('y1', yBot);
      transExtBot.setAttribute('x2', xBot - Math.cos(rad) * extLen);
      transExtBot.setAttribute('y2', yBot + Math.sin(rad) * extLen);

      pTop.setAttribute('cx', xTop);
      pTop.setAttribute('cy', yTop);
      pBot.setAttribute('cx', xBot);
      pBot.setAttribute('cy', yBot);

      // Top arc (inside Z: between left horizontal ray and downward transversal)
      // Left ray is at angle Math.PI. Downward-left transversal is at angle (Math.PI - rad) in standard cartesian,
      // but in SVG Y-down, left ray is at 180 deg, transversal goes to (xBot, yBot) which is at (180 - deg) or (180 + deg)?
      // xTop to xBot: dx is negative (-2*dx), dy is positive (+120).
      // Screen angle: atan2(120, -2*dx) = 180 - deg.
      // Arc sweeps from (180 - deg) to 180.
      const startAngleTop = Math.PI - rad;
      const endAngleTop = Math.PI;

      const pTopArcStart = {
        x: xTop + arcR * Math.cos(startAngleTop),
        y: yTop + arcR * Math.sin(startAngleTop)
      };
      const pTopArcEnd = {
        x: xTop + arcR * Math.cos(endAngleTop),
        y: yTop + arcR * Math.sin(endAngleTop)
      };

      const topD = `M ${pTopArcStart.x} ${pTopArcStart.y} A ${arcR} ${arcR} 0 0 1 ${pTopArcEnd.x} ${pTopArcEnd.y}`;
      topArc.setAttribute('d', topD);
      topArcFill.setAttribute('d', `M ${xTop} ${yTop} L ${pTopArcStart.x} ${pTopArcStart.y} A ${arcR} ${arcR} 0 0 1 ${pTopArcEnd.x} ${pTopArcEnd.y} Z`);

      // Bottom arc (inside Z: between right horizontal ray at 0 deg and upward-right transversal)
      // xBot to xTop: dx is positive (+2*dx), dy is negative (-120).
      // Screen angle: atan2(-120, 2*dx) = -rad.
      // Arc sweeps from -rad to 0.
      const startAngleBot = -rad;
      const endAngleBot = 0;

      const pBotArcStart = {
        x: xBot + arcR * Math.cos(startAngleBot),
        y: yBot + arcR * Math.sin(startAngleBot)
      };
      const pBotArcEnd = {
        x: xBot + arcR * Math.cos(endAngleBot),
        y: yBot + arcR * Math.sin(endAngleBot)
      };

      const botD = `M ${pBotArcStart.x} ${pBotArcStart.y} A ${arcR} ${arcR} 0 0 1 ${pBotArcEnd.x} ${pBotArcEnd.y}`;
      botArc.setAttribute('d', botD);
      botArcFill.setAttribute('d', `M ${xBot} ${yBot} L ${pBotArcStart.x} ${pBotArcStart.y} A ${arcR} ${arcR} 0 0 1 ${pBotArcEnd.x} ${pBotArcEnd.y} Z`);

      // Label positions
      const labelR = arcR + 18;
      const midTop = Math.PI - (rad / 2);
      topLabel.setAttribute('x', xTop + labelR * Math.cos(midTop));
      topLabel.setAttribute('y', yTop + labelR * Math.sin(midTop));

      const midBot = - (rad / 2);
      botLabel.setAttribute('x', xBot + labelR * Math.cos(midBot));
      botLabel.setAttribute('y', yBot + labelR * Math.sin(midBot));

      angleEq.textContent = `${Math.round(deg)}° = ${Math.round(deg)}°`;
    }

    slider.addEventListener('input', (e) => {
      update(parseFloat(e.target.value));
    });

    update(parseFloat(slider.value));
  </script>
</body>
</html>

Angle Pairs Review: Complementary, Supplementary, and Vertical Angles

Angle Properties on Straight Lines

Angle Properties of Parallel Lines and Transversals

Parallel Lines and Transversals: Alternate Interior and Exterior Angles

Using Transversal Angle Relationships to Find Missing Angles

Proving Lines Are Parallel Using Angle Relationships

Constructing the Perpendicular Bisector of a Line Segment

Constructing an Angle Bisector with Compass and Straightedge

Constructing a Perpendicular from a Point to a Line

Constructing a Perpendicular from an External Point

Constructing Parallel Lines Through an External Point

Constructing Regular Polygons: Equilateral Triangle and Square

Using Constructions to Prove Geometric Properties

math

Lines, Angles, and Geometric Constructions

台形の定義と特徴 — 向かい合う1組の辺が平行な四角形

平行四辺形の定義と辺の性質 — 2組の向かい合う辺が平行で等しい

平行四辺形の角の性質 — 向かい合う角の大きさと隣り合う角の和

ひし形の定義と性質 — 4つの辺の長さがすべて等しい四角形

四角形の包含関係と体系的整理 — 長方形・正方形とのつながり

四角形のなかま — 台形・平行四辺形・ひし形の性質

Conditions for Proving a Quadrilateral is a Parallelogram

Special Parallelograms: Rectangles, Rhombi, and Squares

Properties and Proofs of Trapezoids and Kites

Combining Parallel Line Rules with Triangles and Quadrilaterals

Interior and Exterior Angles of Regular Polygons

Regular and Semi-Regular Tessellations of the Plane

Properties and Proofs of Quadrilaterals and Polygons

多边形的概念：凸多边形、顶点、边与正多边形特征

多边形的对角线：从一个顶点引出的对角线与总条数公式

多边形内角和公式：(n-2)×180° 的转化与严格证明

多边形外角和定理：任意多边形外角和恒等于 360°

正多边形的角计算与平面密铺（镶嵌）的数学原理

综合计算：截角问题、漏加/多加一个内角的逆向方程求解

多边形的内角和、外角和与对角线探索

认识平行四边形：定义、几何表示与中心对称性

平行四边形的性质定理（一）：对边相等与对角相等

平行四边形的性质定理（二）：对角线互相平分

平行线间的距离：垂线段的概念、处处相等与度量方法

平行四边形判定定理（一）：一组对边平行且相等

平行四边形判定定理（二）：两组对边分别相等

平行四边形判定定理（三）：对角线互相平分

综合应用：平行四边形判定的选择策略与动态平移几何题

平行四边形的性质、判定与平行线间的距离

矩形的性质定理（一）：有一个角是直角与四个角都是直角

矩形的性质定理（二）：对角线相等与轴对称性

矩形的判定定理：三个角是直角或对角线相等的平行四边形

菱形的性质定理（一）：邻边相等与四条边都相等

菱形的性质定理（二）：对角线互相垂直平分且平分内角及面积公式

菱形的判定定理：四边相等或对角线互相垂直的平行四边形

正方形的定义与性质：既是矩形又是菱形的完美对称图形

正方形的判定定理：从平行四边形、矩形、菱形出发的判定路线

从属关系与中点四边形：任意四边形与特殊四边形的中点连线规律

特殊平行四边形：矩形、菱形与正方形

三角形中位线的概念：中点连线与中线的本质区别

三角形中位线定理：平行于第三边且等于第三边的一半

中位线定理的应用：测量不可达距离与四等分三角形面积

四边形的不稳定性：伸缩拉闸门与三角形斜撑加固原理

认识梯形：底、腰、高及直角梯形、等腰梯形特征与平移辅助线

梯形、四边形不稳定性与三角形中位线定理

四角形の内角の和（360度）と敷き詰めの原理

平行四辺形による平面の敷き詰め活動

ひし形による平面の敷き詰めと美しい模様づくり

台形による平面の敷き詰め活動

一般のどんな四角形でも敷き詰められるかの探究

敷き詰め模様から平行線と角の性質を見つける

タイルの敷き詰めやデザイン（エッシャー風アート）への応用

OGM.10.GEO.3

Aligned learning resources

Related diagrams